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Multiscale mass-spring models of carbon nanotube foams F. Fraternali* Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy, and Division of Engineering, King’s College London, Strand, London WC2R 2LS, UK

T. Blesgen Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany

A. Amendola Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy

C. Daraio Graduate Aerospace Laboratories (GALCIT) and Applied Physics, Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA 91125, USA

Abstract This article is concerned with the mechanical properties of dense, vertically aligned carbon nanotube foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition), which represents a finite portion of the foam thickness. An upper Email addresses: [email protected] (F. Fraternali* ), [email protected] (T. Blesgen), [email protected] (A. Amendola), [email protected] (C. Daraio) *Corresponding author Preprint submitted to Journal Mech. Phys. Solids

September 6, 2010

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Multiscale mass-spring models of carbon nanotube foams

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This article is concerned with the mechanical properties of dense, vertically aligned carbon nanotube foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where in nitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation a non-homogeneous buckling regime, and a densi cation response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain de nes a mesoscopic dissipative element (micro-meso transition), which represents a nite portion of the foam thickness. An upper scale model formed by a chain of nonuniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are tted with excellent agreement. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: a. REPORT

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Graphical Abstract

Graphical abstract

Σ@MPaD 25

1st cycle, 5 springs experimental

20

numerical 15

(movie) 10

5

0

Ε 0

Axial strain localization in a mesoscopic chain of five bistable springs. The spring collapse mimics the local kinking of compressed carbon nanotubes.

0.2

0.4

0.6

0.8

Predicted stress-strain response (solid line) at the macroscopic scale, reproducing the experimental behavior of a real CNT foam (dashed line).

Research Highlights

Research highlights • Axial strain localization in microscopic bistable spring chains mimics kinking of compressed carbon nanotube arrays. • Infinitesimal viscous events at the microscale induce time-independent hysteresis at the mesoscale. • Multiscale mechanical modeling of CNT foams is obtained through an informationpassing approach. • Available experimental results on compressed CNT foams are reproduced with excellent agreement.

1

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Multiscale mass-spring models of carbon nanotube foams F. Fraternali* Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy, and Division of Engineering, King’s College London, Strand, London WC2R 2LS, UK

T. Blesgen Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany

A. Amendola Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy

C. Daraio Graduate Aerospace Laboratories (GALCIT) and Applied Physics, Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA 91125, USA

Abstract This article is concerned with the mechanical properties of dense, vertically aligned carbon nanotube foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition), which represents a finite portion of the foam thickness. An upper Email addresses: [email protected] (F. Fraternali* ), [email protected] (T. Blesgen), [email protected] (A. Amendola), [email protected] (C. Daraio) *Corresponding author Preprint submitted to Journal Mech. Phys. Solids

September 6, 2010

scale model formed by a chain of nonuniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are fitted with excellent agreement. Key words: Carbon nanotube foams, bi-stable springs, multiscale behavior, strain localization, hysteresis 1. Introduction Since their discovery (Radushkevich and Lukyanovich, 1952; Oberlin et al., 1976; Iijima et al., 1995), carbon nanotubes (CNTs) have been widely studied for the understanding of different aspects of their chemical, electrical and mechanical responses. Because of their unique properties and their multiscale nature, forests of vertically aligned carpets of CNTs have been proposed for a variety of applications (Gjerde et al., 1991; Veedu et al., 2006; Daraio et al., 2004b; Majumder et al., 2005; Maheshwari and Saraf, 2008). The mechanical response of individual nanotubes under axial and radial deformation, and their bending/buckling modes, have been studied extensively using experimental, theoretical and molecular-dynamics analysis (Iijima et al., 1995; Yakobson et al., 1996; Falvo et al., 1997; Belytschko et al., 2002; Arroyo and Belytschko, 2003; Pantano et al., 2004; Cao and Chen, 2006). The elastic modulus E of individual carbon nanotubes has been reported to be very high: ∼ 1 TPa (Pantano et al., 2004). However in experiments this value can vary widely, depending on the number of defects, the CNT microstructure and the synthesis method followed. The study of the mechanical properties of CNTs was later extended to bundles of nanotubes under pressure (Chesnokov et al., 1999; Peters et al., 2000; Chan et al., 2003; Liu et al., 2005; Qi et al., 2003) and to CNT forests under nanoindentation (Mesarovic et al., 2007). The study of thin structural foams (Gibson and Ashby, 1998) for cushioning (Zhang et al., 2009), energy dissipation (Teo et al., 2007) and protection (Liu et al., 2008) has recently received increasing attention for several practical applications, including mitigation of explosive loading (Nesterenko, 2001). Nanotube based films have been reported as an excellent alternative to regular foams, exhibiting a supercompressible foam-like behavior under compressive cycling loads (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Cao et al., 2005). 2

Investigations on the dynamic response of foam-like forests of CNTs under dynamic ball impacts have also been performed (Daraio et al., 2004b; Daraio et al., 2004b; Daraio et al., 2006; Misra et al., 2009). Results show a strongly nonlinear response that appears to be very suitable for energy-absorbing layered materials in noise and shock wave mitigation and as nonlinear springs for assembling nonlinear acoustic crystals. The CNT forests’ response was also found to be strongly dependent on the forests’ microstructure (height, density, alignment, etc.) and growth method. In certain cases the possible presence of plastic deformation and fracturing of the tubes was reported (Daraio et al., 2004b). Mechanical models consisting of chains of identical bi-stable springs have been extensively studied by several authors, since the can describe a series of relevant nonlinear material behaviors (e.g., phase transformations, reversible pseudo-plasticity, hysteresis, fracture), through the interplay between macroscopic and microscopic length scales (refer e.g. to Ortiz, 1999; Puglisi and Truskinovsky, 2000, 2002, 2005). It is well known that such systems exhibit a ‘bumpy’ multi-well energy landscape, allowing for multiple metastable equilibria (cf., e.g., Blesgen, 2007; Braides and Cicalese, 2007; Charlotte and Truskinovsky, 2002, 2008; Ortiz, 1999; Puglisi and Truskinovsky, 2000, 2002). In particular, Pampolini and Del Piero (2008) have recently found that they well describe the hysteretic response of open-cell polyurethane foams under confined compression tests. In this article, we present a phenomenological model of the mechanical response of carbon nanotube foams under compressive loading, which is inspired by some distinctive features of the micromechanical response reported earlier (cf., e.g., Cao et al., 2005; Zbib et al., 2008; Misra et al., 2009; Hutchens et al., 2010). The given model makes use of multiscale chains of lumped masses connected by nonlinear springs. It captures the ‘three-phase’ compressive deformation response of CNT forests shown by a number of experimental studies. The compressive deformation response is characterized by an initial elastic deformation, a non-homogeneous buckling (or plateau) regime, often featuring a sawtooth-like profile, and a densification phase. This three-phase response, which is common for cellular materials (Gibson and Ashby, 1998), is usually accompanied by marked hysteresis and strain localization in CNT structures. We show in Section 3 that a series of bistable elastic springs (Fig. 1) described by the potential in Eq. (1) exhibits a similar stress-strain response (Fig. 6), and through-the-thickness localization of the axial deformation (Fig. 7). The latter effect mimics the snap-buckling 3

events observed through Scanning Electron Microscope (SEM) in real CNT arrays (cf. Fig. 2 of Cao et al., 2005; Fig. 3 of Hutchens et al., 2010). Such a model therefore appears to be effective in describing the microstructure rearrangements taking place in compressed CNT foams, within a simple 1D framework. We focus on the time-independent component of the hysteresis associated with the compressive loading/unloading of CNT foams, which is essentially due to effects such as friction, entanglement and electrostatic interaction between individual and bundles of carbon nanotubes (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Misra et al., 2009; Cao et al., 2005). The macroscopic hysteresis is described as a rate-independent phenomenon induced by the succession of infinitesimal viscous events at the microscopic scale (Puglisi and Truskinovsky, 2005). As a result, the mechanical model presented in this work does not account for viscosity or other rate-dependent effects at the macroscopic scale, which we address to future work. Most of the available studies on bistable spring chains consider only two spatial-temporal micro-macro scales. The formulation adopted in the present study instead introduces three different time-space scales: a ‘microscopic’ scale (of the order of nanometers), which is associated with the individual bi-stable springs and an infinitesimal portion of the total foam thickness Ltot ; a ‘mesoscopic scale’ (of the order of micrometers), corresponding to the limit of an infinite series of microscopic springs and representing a finite portion of Ltot ; and a ‘macroscopic’ scale (of the order of millimeters), describing the entire foam. We start with the derivation of a discrete 1D model at the microscopic level, showing N + 1 particles having nearest neighbors connected by N uniform nonlinear springs. The discrete spring potentials are chosen to allow the study of softening and strain localization of the tubes. This is discussed in Section 2. Here, we build on the concept of bi-stable springs discussed by Puglisi and Truskinovsky in (Puglisi and Truskinovsky, 2000; Puglisi and Truskinovsky, 2002; Puglisi and Truskinovsky, 2005;). In Section 3 we determine analytically the continuum limit N → ∞ of the microscopic spring chain, by particularizing to the present case the analysis presented in Puglisi and Truskinovsky (2005). Such a limit defines a mesoscopic dissipative element. In Section 4 we then formulate an upper scale model of the entire CNT foam through an information-passing approach, by superimposing a finite number of mesoscopic dissipative springs with different mechanical properties. We account in this way for inhomogeneities 4

induced by the CNT growth process, which several authors think be a leading cause of the discrete folding/buckling events described above (Cao et al., 2005; Hutchens et al., 2010). The introduction of non-uniform mechanical properties allows us to enrich the formulation given in Puglisi and Truskinovsky (2000), Puglisi and Truskinovsky (2002), Puglisi and Truskinovsky (2005), Pampolini and Del Piero, (2008) for bistable spring chains, modeling macroscopic hardening, instead of a perfectly ‘plastic’ response. Hardeningtype post-buckling regimes are commonly observed in compression tests on CNT foams (Cao et al., 2005; Misra et al., 2009). We present in Section 5 a numerical micro-meso convergence study, and the fitting of mesoscopic model parameters to different available experimental data on compressed CNT foams. We demonstrate that the proposed model and its numerical implementation are capable of reproducing the physical behavior of real CNT foams with excellent agreement. We end the article with a critical evaluation and discussion of the results and an outlook. 2. The mechanical model at the microscopic scale We model an infinitesimal portion of a CNT array as a collection of N + 1 lumped masses m0 , . . . , mN piled up one over the other (with N > 2). In this configuration, m0 is clamped at the bottom of the pile at the position x0 = 0, whereas mN is on top at position xN = L > 0, referring to the unstressed reference configuration (Fig. 1c). The nearest neighboring mass points are connected by N nonlinear microscopic springs. We assume that the reference configuration displays equal distances hN := L/N between the masses, and denote the axial displacement of the mass mi (positive upward) by uiN (with u0N = 0, see Fig. 1c). We set uN := {u0N , ..., uN N }. i For the mechanical energy V of the microscopic spring placed between nearest neighbors mi−1 and mi , we assume the three-branch expression defined by i i Va (ε ) = −k0i [εi +ln(1−εi )], εi < εia , i i Vbi (εi ) = c1 + σai εi + 12 kbi (εi −εia )2 , εia ≤ εi ≤ ε¯ic , V (ε ) = i i Vc (ε ) = c2 −kci [εi −εi∗ +ln(1−(εi −εi∗ ))], ε¯ic < εi ,

5

(1)

r Figure 1: (Color online) (a) Schematic diagram of a vertically aligned CNTs foam, uniformly loaded in compression. (b) SEM of the as grown carbon nanotube film showing the alignment and the microstructural layering due to the growth process (Deck and Vecchio, 2005). (c) Modeling of a portion of a CNT foam as a collection of microscopic mass-spring elements.

with εi = εi − εi0 , where i

i

ε = ε (uN ) =

i ui−1 N − uN hN

(2)

is the strain measure associated with such a spring (positive in compression). In (1), the quantity εi0 ≥ 0 determines the value of εi corresponding to the first minimum of V i (‘equilibrium’ or ‘initial’ strain); k0i > 0, kbi < 0, kci > 0, εia > 0 and εic ≥ εia are constitutive parameters (five independent parameters); the constants c1 < 0 and c2 > 0 are such that Vai (εia ) = Vbi (εia ), Vbi (¯ εic ) = Vci (¯ εic ); and it results

6

εi∗

=

ε¯ic =

εic

σai − i , kc + σai

(3)

εic (kci + σai ) (σci − σai ) (kci + εic kci + εic σai ) , + kci + σci (kci + σai ) (kci + σci )

(4)

with

σai = k0i

εia , 1 − εia

σci = σai + kbi (¯ εic − εia ).

(5)

The relationship

σ i (εi ) = V i

0

εi εi < εia , k0i 1−ε i, σai + kbi (εi − εia ), εia ≤ εi ≤ ε¯ic , = kci (εi −εi∗ ) , ε¯ic < εi 1−(εi −εi ) ∗

defines the stress σ i acting in the spring connecting mi−1 with mi .

7

(6)

We deduce from (1) that V i owes the two-well profile shown in Fig. 2. Accordingly, the stress-stain relationship (6) is described by the non-monotone profile depicted in Fig. 3. It is worth noting that the mechanical response of the generic microscopic spring encompasses two stable phases, for εi < εia (phase a) and εi > ε¯ic (phase c), respectively; and an intermediate unstable phase b for εia ≥ εi ≥ ε¯ic (spinodal regime). For future use, we set ∆σ i := σci − σai ,

ε¯ia :=

εia σci σai + εia ∆σ i

(7)

i denote the slope of the linear branch of the convex hull of V i and let σM (Maxwell stress, see Fig. 3). We also convene to denote the value of εi where i i in phase c σ i = σM in phase a by εiaM , and the value of εi such that σ i = σM i by εcM . With reference to the generic spring, Equation (6)1 highlights that k0i represents the elastic stiffness at zero stress in phase a (εi = 0). On the other hand, Equation (6)2 points that kbi represents the (constant) stiffness in the spinodal phase b, while Equation (6)3 reveals that kci represents the stiffness at zero stress in phase c (εi = εi∗ ). We name ‘symmetric’ the case with kci = k0i and ‘asymmetric’ the one with kci 6= k0i . The meaning of the other quantities appearing in Equations (1) - (7) is illustrated in Figs. 2 and 3.

3. Dynamic relaxation and hysteresis at the mesoscale There are two fundamental characteristics reported experimentally in the bulk compressive response of dense vertically aligned CNTs forests: (i) a good recovery of deformation even at large compressive strains after a sufficiently large recovery time (Cao et al., 2005; Misra et al., 2009); and (ii) a strong hysteresis, observed in particular in as-grown foams, attributed to several potential effects, including friction, entanglement and electrostatic interaction between individual and bundles of carbon nanotubes (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Misra et al., 2009; Cao et al., 2005). In the present Section we will analyze such a dissipative behavior, by studying a dynamic switching process at the microscopic scale between the phases (a) and (c) described in Figs. 2 and 3. This is in line with the ideas in Puglisi and Truskinovsky (2002), Puglisi and Truskinovsky (2005). Following Puglisi and Truskinovsky (2005), we name a response of the material plastic, if the strain εi of a single spring exceeds εi0 + εia . For 8

Vi

C2

C1 Mc

A1 A2 Εia

Ma ΕiaM Εia

Εic ΕicMΕic

Εi =Εi -Εi0

Figure 2: (Color online) Mechanical energy V i of the generic microscopic (bi-stable) spring.

a chain of N springs, this can be characterized by the occurrence of loading and unloading stress plateaux. However, we notice that our analysis excludes accumulation of permanent deformation, therefore the end point of one hysteresis cycle coincides with the start point of the next cycle. In this sense, the present hysteresis model is time-independent. We will show in Section 5.2 that it is nevertheless capable of capturing a previously accumulated permanent deformation (mechanical preconditioning), through suitable definition of the initial strains εi0 . Within the current Section, we rescale for simplicity L to unity, and assume that V i is independent of the spatial position. Accordingly, we drop the superscript i in front of the spring properties. This because we refer the following analysis to a finite portion of the CNT foam, regarding such a mesoscopic element as the limit for N → ∞ of a series of N identical microscopic springs. Furthermore, we restrict our attention to the case with kc = k0 (”symmetric” model), and ε0 = 0. For the present analysis we 9

Σi

C1

A1 Σai Ma A2

Mc C2

Σ1i =ΣMi

Σci Εia

ΕiaM Εia

Εic ΕicMΕic

Εi =Εi -Εi0

Figure 3: (Color online) Stress σ i versus strain εi relationship in the generic microscopic spring.

require a certain smallness condition on εa and εc relating to strong pinning that disappears in the limit N → ∞, see Puglisi and Truskinovsky (2002). We define the mechanical energy of the foam as N 1 X EN (uN ) = V (εi (uN )) N i=1

(8)

with the effective potential if ε < εa , −k0 [ε + ln(1 − ε)] kb 2 V (ε) = c + σa ε + 2 (ε − εa ) if εa ≤ ε ≤ εc , 1 c2 − k0 [ε − ε∗ + ln(1 − ε + ε∗ )] if εc < ε. Let σ be the given total stress, coinciding at equilibrium with the stress in each individual spring (σ = σ 1 = . . . = σ N ). The total average strain is 10

simply N 1 X i ε (uN ). ε(uN ) := N i=1

We model plasticity by the gradient flow equations (Puglisi and Truskinovsky, 2005) ∂ΦN ν ε˙i (uN ) = − i (ε1 (uN ), . . . , εN (uN )) (9) ∂ε with the total energy N 1 X ΦN (ε , . . . , ε ) := V (εi ) − σεi . N i=1 1

N

The evolution equation (9) lets εi evolve towards local minimizers of ΦN . We are interested in the limit ν → 0 which amounts to infinitely fast evolution such that ε(uN ) attains a local minimizer of ΦN . First we construct the equilibrium points. Inside the i-th spring element, the strain must satisfy the condition V 0 (εi ) = σ. For given total stress σ, there are at most the three local minimizers ε˘a =

σ , k0 + σ

ε˘b =

σ − σa + εa , kb

ε˘c =

σ(1 + ε∗ ) + k0 ε∗ = ε˘a + ε∗ . k0 + σ

Let p, q, 1 − p − q denote the phase fractions of the minimizers a, b, and c, which corresponds to having N p, N q, N (1 − p − q) springs in phase a, b, and c, respectively. As ε 7→ V (ε) is concave in Regime b, if the elongation of a spring in the local minimum ε˘b is altered by an arbitrarily small perturbation, it will move (according to the sign of the perturbation) to either ε˘a or ε˘c . In consequence, any system of N springs with q 6= 0 is unstable and we may in the following restrict to the case q = 0. From ε = p˘ εa +(1−p)˘ εc we compute the equilibrium stress-strain relation to be k0 (ε − εp ) (10) σ(ε) = 1 − (ε − εp ) with εp := (1 − p)ε∗ that can in a natural way be identified with the plastic strain. From (10) we see that σ only depends on the elastic strain εel := ε − εp . 11

For the energy of the equilibrium configuration with N p springs in phase a we find i h σ(ε) k0 bp (ε) = −k0 + ln + (1 − p)c2 E k0 + σ(ε) k0 + σ(ε) = −k0 (ε − εp ) + ln(1 − (ε − εp )) + (1 − p)c2 . bp has a finite number of local minimizers (depending on the Note that E remaining parameter p). As explained in Puglisi and Truskinovsky (2005), the switching takes place between branches that differ in exactly one element in the phase state and the succession of N such steps describes the transition from one homogeneous state to the next. Each of these steps can be thought of as the combination of an elastic part and a plastic part. The stress-strain curve of the foam follows a sawtooth pattern as illustrated in Fig. 4 (left). We denote by A¯i the end point of the i-th branch. ¯ i → A¯i+1 ) where the system The A¯i are the final states of the elastic steps (B remains on the same metastable branch as long as possible. The plastic steps ¯ i ) are characterized by that the total strain is fixed and the system (A¯i → B switches between metastable branches that are neighbors ([N p] = 1, and [Q] generically denotes the jump of a quantity Q). These considerations lead to the representation k0 εa i ¯ , A = εA¯i , 1 − εa

k0 (εa − ε∗ /N ) i ¯ B = εA¯i , , 1 + ε∗ /N − εa

1≤i≤N

in the (ε, σ)-diagram, where i−1 ε∗ . N

εA¯i := εa +

So we can compute that a plastic step is characterized by [ε] = 0,

[N p] = 1,

[σ] = −

k 0 ε∗ , N (1 − εa )(1 + ε∗ /N − εa )

whereas an elastic step fulfills [ε] =

ε∗ , N

[N p] = 0,

[σ] = +

12

k0 ε∗ . N (1 − εa )(1 + ε∗ /N − εa )

(11)

Σ

Σ

A

1

A

2

B

1

B

2

B2 B1 A2 A1 Ε

Ε

Figure 4: Overall stress-stain pattern for a chain with N = 10 microscopic springs (left), and limit response for N → ∞ (right).

The evolution equation (9) lets εi evolve towards local minimizers of ΦN . Now we want to look at the energetics of the plastic and the elastic regime. For an elastic step we have the energy difference bN = k0 ln(1 − εa ) − ln(1 − εa + ε∗ /N ) + k0 ε∗ + c2 . ∆E N In the same spirit, we calculate that for a system with N ≥ 1 springs, the plastic dissipation is DN

i c +k ε 2 0 ∗ := k0 ln(1 − εa + ε∗ /N ) − ln(1 − εa ) − N 1 ε ∗ c2 + k 0 ε ∗ = k0 − for a ξN ∈ (1 − εa , 1 − εa + ε∗ /N ). (12) ξN N N h

Clearly, ξN → 1 − εa for N → ∞. In one hysteresis cycle, there are N loading steps and N steps when the system is unloaded, so we have totally 2N steps that dissipate energy. The total dissipated energy D in a cycle becomes in the limit N → ∞ 2k ε ε 0 ∗ a D = lim 2N DN = − 2c2 . (13) N →∞ 1 − εa 13

We put the modulus here to ensure that the dissipation is positive. The limit stress-stain pattern for N → ∞ is shown in Fig 4. It corresponds to a ”perfectly plastic” behavior with stress plateaux at σ = σa = k0 εa /(1 − εa ) (loading plateau) and σ = σc = σa + ∆σ (unloading plateau). We emphasize again that this ansatz only works for rate-independent plasticity where the energy only depends on start point and end point of the evolution, but not on the evolution path itself. The limit dissipation (13) equals the area enclosed by the limit stress-strain response. As already observed, we confer the behavior shown in Fig. 4 (right) to a mesoscopic spring element, which represents a finite portion of the CNT foam thickness. 4. Multiscale numerical modeling We formulate in the present Section a multiscale numerical model of a nonlinear mass-spring chain, where each spring represents either a microscopic bistable element (cf. Section 2), or a mesoscopic dissipative element of the kind introduced in the previous Section. We introduce two different time scales: an external (slow) time τ ∈ [τ0 , τ1 ] ruling an evolution law of the applied boundary conditions, and an internal (fast) time t ∈ [t0 , t1 ] governing the dynamic relaxation of the system for fixed τ . Depending on the adopted model for the individual springs, we may have the t corresponds to the microscopic timescale (i.e. to the time ruling the microscopic behavior) and τ to the mesoscopic time (micro-meso transition), or, alternatively, that t corresponds to the mesocopic time and τ to the macroscopic time (meso-macro transition). Let us denote a prescribed displacement time-history of the topmost mass N m by δ(τ ). We introduce a discretization {τ1 , . . . , τM } of the loading interval [τ0 , τ1 ] and compute the system response for fixed τ = τk and δ¯ = δ(τk ), through integration with respect to t of the evolution equations mi u¨ˆiN + γ i uˆ˙ iN = σ i+1 − σ i ,

i = 1, . . . , N,

(14)

which generalize the gradient flow equations (9) (Puglisi and Truskinovsky, 2005). In (14), uˆiN = uˆiN (t) denote transient displacement histories of the masses m0 , . . . , mN at fixed τ ; σ i indicates the current stress in the ith spring (it is understood that it results σ i = 0 for i > N ); and γ 1 , . . . , γ N denote damping coefficients. Since we are only interested in the final equilibrium configuration of the transient internal motion, we ‘overdamp’ such a motion down, by introducing 14

fictitious masses and damping factors in (14). In detail, we set the integration i time step ∆t to unity, and introduce fictitious masses mi = αk ∆t2 , with p ki = hN (V i + V i+1 )00 and α ≥ 100. This ensures ∆t ≤ 0.1 mi /k i (i = 1, ..., N ) (Fraternali et al., 2009). Moreover, we let the generic γ i be equal to the ‘critical’ damping defined as follows √ (15) γ i = 2 mi k i The equations of motion (14) are associated with the initial conditions uˆiN (t = t0 ) = (uiN )(k−1) , i = 0, . . . , N − 1; uˆ˙ iN (t = t0 ) = 0,

¯ uˆN N (t = t0 ) = δ

i = 0, . . . , N

(16)

where (uiN )(k−1) (i = 1, . . . , N ) are the displacements of the masses at the external time τ = τk−1 . Equations (14), (16) are numerically integrated through a fourth-order Runge-Kutta integration scheme, up to an internal time t1 such that it results |σ i+1 − σ i | ≤ 10−6 |σ N | for all i ∈ {1, . . . , N − 1}. When the internal equilibrium is reached, we set k ← k + 1 and re-iterate problem (14). For the micro-meso convergence study of Section 5.1, we consider the stress-strain law described by Equation (6) and illustrated in Fig. 3, for each microscopic spring. For the simulation of compression tests on real CNT foams, we instead adopt nonuniform chains of mesoscopic spring characterized by a suitable numerical regularization of the stress-strain pattern shown in Fig. 4. In detail, we introduce a ‘hardening’ type regularization consisting of the following stress-strain law (Fig. 5)

i

σ =

k0i εi σ (a,i) = (1−ε i) , i (εi −εia ), σ (d,i) = σai + kh+

(εi < εia ) and (flag(k−1) 6= c), (εia ≤ εi ≤ εˆic ) and (flag(k−1) = a),

i εia ≤ εi ≤ ε¯ic ) and (flag(k−1) = c), σ (e,i) = σai + ∆σ i + kh− (εi − ε¯ic ), (ˆ kci (εi −εi∗ ) σ (c,i) = 1−(ε (¯ εic < εi ) and (flag(k−1) 6= a), i −εi ) , ∗ (17) 15

where, for each τ = τk (k = 1, . . . , M ), we set a, (εi < εia ) and (flag(k−1) 6= c), c, (¯ εic < εi ) and (flag(k−1) 6= a), = flag(k−1) , ((εia ≤ εi ≤ εˆic ) and (flag(k−1) = a)), or ((ˆ εia ≤ εi ≤ ε¯ic ) and (flag(k−1) = c)).

flag(k)

(18)

i i and kh− in (17) are constitutive The quantities k0i , ∆σ i , kci , εai , εic , kh+ parameters (seven independent parameters), while εˆia and εˆic are computed by solving for εi the equations

σ (a,i) = σ (e,i) ,

σ (c,i) = σ (d,i)

(19)

respectively. One gets back to a perfectly plastic mesoscopic response by i i setting the regularization (‘hardening’) parameters kh+ and kh− equal to i zero. The quantities ε are given by (2) under the replacement of uiN with uˆiN .

Σi C1

A1 Σai Ma

Mc

ΣMi Σci

A2

C2

`i Εa

Εic

ΕiaM Εia

`i ΕicM Εc

Εi =Εi -Εi0

Figure 5: (Color online) Hardening-type regularization of the mesoscopic response.

16

5. Applications The present Section deals with applications of the above numerical model to a convergence study and the fitting of laboratory tests on the cyclic compression of CNT foams. In the first case, series of springs with constitutive equations (6) were considered (model # 1: micro-meso transition), letting the total number of springs of the system increase progressively (micro-meso transition). In the second case, spring chains with constitutive equations (17) were instead examined (model # 2: meso-macro transition). The optimal fit of available experimental stress-strain curves was performed by playing with the constitutive parameters of equations (17) and the number of mesoscopic springs N . Depending on the value of N , a significant number of parameters may need to be identified. This lead us to employ a fitting procedure based on genetic algorithms, which have been proved to be well-suited for multimodal non-convex optimization (cf. e.g. El Sayed et al., 2008; Fraternali et al., 2009). In all the examined examples, we name global strain the quantity ε = (L − `)/L, ` denoting the total deformed length of the chain. 5.1. Convergence study and micromechanics of ‘plastic’ steps We examined uniform chains with fixed length L = 860 µm and increasing number of microscopic springs N , subject to a complete loading-unloading compression cycle up to a global strain ε = 0.85 (as in the experiments examined in Section 5.2.2). We employed model #1 with kc = k0 (‘symmetric’ case); ε0 = 0; and the material properties listed in Tab. 1 of the Appendix for all the springs. Fig. 6 shows the numerically computed dissipation DN versus strain ε and stress σ versus ε responses of such a model for different numbers of springs (N = 5, 10, 15 and 50). The σ-ε plot in Fig. 6 highlights that the global stress-strain response alternates elastic steps and ‘plastic’ jumps of σ at constant ε, oscillating converging to the ‘perfectly plastic’ mesoscopic behavior shown in Fig. 4. The DN -ε plot in Fig. 6 instead shows that DN converges to the limit dissipation defined in equation (13). The above results therefore confirms the theoretical predictions of Section 3. Selected equilibrium configurations and a deformation animation (online version only) of the model with N = 5 springs are given in Fig. 7. One can easily recognize the configurations corresponding to the plastic steps of the microscopic chain, displaying the snap of a single spring and the simultaneous elastic rearrangement of the remaining springs (cf., e.g., configurations 3 and 5 from the left). It is worth noting that the ordering of the springs is of no relevance in the 17

present case. Therefore, the succession of the spring snaps can indifferently proceed from the bottom to the top of the chain, as shown in Fig. 7, from top to bottom, or in random sequence. In a real-world chain of springs, the parameters as k0 or εa would never be perfectly identical and the small deviations of the different springs would break the symmetry and determine the ordering when the springs snap. 5.2. Fitting of experimental results on compressed CNT foams We examined the experimental results of a cyclic compression test on a doubly anchored CNT foam run at the Graduate Aerospace Laboratories of the California Institute of Technology, and the results given in Cao et al. (2005) on the cyclic compression of a foam-like CNT film. We fit model # 2 both to he first loading/unloading cycle of the examined foams, and to selected cycles following an initial mechanical preconditioning, employing the Breeder Genetic Algorithm (BGA) presented in the Appendix. For the fit of the first cycles we assumed εi0 = 0 in each spring, while for the cycles following the foam preconditioning we set εi0 = ε0 in each spring, ε0 being the macroscopic (permanent or transient) strain observed at zero stress at the end of the previous loading cycle. For the cycles following the preconditioning, we considered two different fitting models: one accounting for hysteresis (numerical (1): ∆σ i < 0 in each mesoscopic spring), and the other without hysteresis (numerical (2): ∆σ i = 0 in each mesoscopic spring). Since all the adopted fitting models account for macroscopic time-independent behavior (cf. Sect. 3), our fitting results are independent of the strain rate actually applied in the experiments. 5.2.1. Compression tests on a doubly anchored CNT foam We performed cyclic compression tests on vertically aligned multi-walled carbon nanotube forests (800µm in length with sample area ∼ 14mm2 ) grown by chemical vapor deposition (CVD) using ferrocene and toluene as precursors. The average diameter of the as-grown CNTs was ∼ 50nm. The as-grown CNTs foams were partially anchored between two polymer layers to provide structural support and sample transportability. For anchoring the CNT foams on a substrate we spin-coated polydimethylsiloxane (PDMS) on top of a glass slide at 800 RPM to get 50 − 100µm thick films. Before curing the polymer we partially embedded the CNT-foams at 80◦ C for 1 hour. After curing, the carbon nanotubes protruding from the substrate showed excellent vertical alignment with an average height of 18

∼ 750µm. The process was repeated turning the sample upside down to obtain a ‘doubly anchored’ system (i.e. sandwich structure with polymer on both sides and the CNT foam in the middle). Typical experimental results obtained from cyclic compression tests are shown in Fig. 8 (top). We separately fit the first and the fourth stress-strain cycles shown in Fig. 8 to model # 2, considering the general ‘asymmetric’ case with kci 6= k0i (i = 0, . . . , N − 1) and a chain with N = 4 springs. For the fourth cycle we accounted for an initial strain ε0 = 0.20, which approximatively corresponds to the strain measured at zero stress at the end of the third loading cycle (Fig. 8, top). The best fit parameters obtained through BGA optimization are given in Tab. 2 of the Appendix. A comparison between best-fit and experimental overall stress-strain curves is shown in Fig. 8 (center and bottom). One observes that the non-uniform dissipative mass-spring model is able to capture the real hysteretical behavior of the examined CNT foam, both at their pristine state (first cycle) and after mechanical preconditioning (fourth cycle), through a multi-plateaux overall stress-strain profile. In particular, Fig. 8-bottom and Tab. 2 of the Appendix show that the response after preconditioning can be roughly described through a suitable non-dissipative model. 5.2.2. Compression tests on a foamlike CNT film We fit a cyclic compression test given in Cao et al. (2005) for a CNT foamlike film with thickness L = 860 µm. The analyzed experiment performed 1000 loading/unloading cycles up to a global strain ε = 0.85 (cf. Fig. 3B of Cao et al., 2005 and Fig. 3 of the Appendix). The fitting of the ‘symmetric’ formulation of model # 2 (kci = k0i in each spring) to the first cycle is presented in the Appendix. A dramatic improvement of the fitting ability of model # 2 was obtained by considering the ‘asymmetric’ case (kci 6= k0i ). As shown in Tab. 4 of the Appendix, we were indeed able to reduce the fitting fitness to 0.808 MPa by using a BGA-optimized ”asymmetric” multiscale model with 10 springs. Such a fitness value is markedly lower than that obtained for the corresponding ”symmetric” case (1.462 MPa, cf. Tab. 3 of the the Appendix). We were able to optimally fit also the 1000th cycle of the examined experiment to the ‘asymmetric’ model, obtaining a fitting fitness of 0.444 MPa through the BGA-optimized 5 spring model described in Tab. 4 of the Appendix. The excellent match between numerical and experimental results for the present case is illustrated by Fig. 9, which compares predicted and measured overall stress-strain responses. For the first cycle 19

we observe that the numerical multi-plateaux response closely approaches the continuous experimental recording as the number of mesoscopic springs increases from 5 to 10 (Fig. 9, top and center). Regarding the 1000th cycle, we assumed ε0 = 0.14 as suggested in Cao et al. (2005). Fig. 9, bottom shows that the experimental stress-strain profile corresponding to the 1000th cycle is already excellently approximated by a 5 spring model with dissipation (numerical (1)), and roughly described by an analogous model without dissipation (numerical (2)). One observes from Tabs. 3 and 4 of the Appendix that the average value of the stiffness k0i , among all the springs, is approximatively equal to the elastic modulus estimated by Cao et al. (2005) for the present CNT foam (≈ 50 MPa). 6. Discussion and Outlook In this article we derived a mechanical model describing the behavior of CNT foams under uniform compression, through the concept of bi-stable springs presented in Puglisi and Truskinovsky (2000, 2002, 2005), which has been recently applied to open-cell polyurethane foams (Pampolini and Del Piero, 2008). The proposed model differs from other bi-stable mass-spring models available in the literature (refer e.g. to Blesgen, 2007; Braides and Cicalese, 2007; Charlotte and Truskinovsky, 2002, 2008; Ortiz, 1999; Puglisi and Truskinovsky, 2000, 2002, 2005, and references therein), due to the presence of an intermediate, mesoscopic scale, placed in between the microscopic scale of the bi-stable springs, and the macroscopic scale of the entire structure. The microscopic scale aims to describe the dynamic snapping of the carbon nanotubes, due to local buckling (Cao et al., 2005; Zbib et al., 2008; Misra et al., 2009; Hutchens et al., 2010), while the mesoscopic scale is intended to describe the time-independent hysteretic behavior of finite portions of the CNT foam. The latter typically follows from kinking, entanglement of the tubes, friction and other microscopic dissipative effects (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Misra et al., 2009). Another relevant difference between the present model and most of the available bi-stable mass-spring models consists of the fact that the current model accounts for nonuniform mesoscopic spring properties, while other models usually consider uniform chains of bi-stable springs (see e.g. Puglisi and Truskinovsky, 2000, 2002; Pampolini and Del Piero, 2008). Such a mechanical inhomogeneity allows us to account for hardening of the macroscopic 20

response in the post-buckling range (Cao et al., 2005; Misra et al., 2009). The numerical results given in Section 5 demonstrated that our methods are capable of recovering available experimental results on the compression of CNT foams with excellent agreement. We fitted the parameters of the mesoscopic springs through a Breeder Genetic Algorithm, which is well suited for global non-convex optimization. The fitting of experimental stress-strain curves allowed us to recognize that non-uniform dissipative mass-spring models are well suited to capture the main features of the real compressive response of CNT foams, and specifically strain localization due to CNT kinking and time-independent hysteresis. The latter was found to be relevant during loading/unloading from the pristine state, and progressively decaying after mechanical preconditioning. By setting to zero the dissipation of the model, we were led to obtain a rough, non-linearly elastic approximation of the examined experimental behaviors after preconditioning. The distinctive features of the multiscale model presented in this work highlight its potential use to describe the mechanical behavior of multilayered systems composed of alternating CNT foams and anchoring polymeric films (Misra et al., 2009). We will address such an extension of the current model in future work. Other future directions of the present research might regard the description of permanent deformation through fatigue or damage mechanisms; the inclusion of rate-dependent dissipative effects at the macroscopic scale; and the modeling of long-range interactions mimicking van der Waals forces. Acknowledgements The authors would like to thank Dr. Abha Misra and Dr. Luigi de Nardo for performing the experiments presented in Section 5.2.1. CD acknowledges support from the Institute for Collaborative Biotechnologies under contract W911NF-09-D-0001 with the Army Research Office. FF greatly acknowledges the support of the University Centre for Risk Prediction and Prevention (CUGRI) in association between the Universities of Salerno and Napoli Federico II, Italy. FF also thanks the Graduate Aerospace Laboratory at the California Institute of Technology (GALCIT) for the hospitality during his visit.

21

Appendix. Supplementary data Supplementary data associated with Section 5 of this article can be found in the online version. References Arroyo, M., Belytschko, T., (2003). A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of nanotubes. Mech. Mater. 35, 193-215. Belytschko, T., Xiao, S.P., Schatz, G.C., Ruoff, R.S., (2002). Atomistic simulations of nanotube fracture. Phys. Rev. B65, 235430. Blesgen, T., (2007). On the competition of elastic energy and surface energy in discrete numerical schemes. Adv. Comput. Math. 27, 179-194. Braides, A., Cicalese, M., (2007). Surface energies in nonconvex discrete systems. Math. Mod. Mrth. Appl. S. 17(7), 985-1037. Cao, A., Dickrell, P.L., Sawyer, W.G., Ghasemi-Neihad, M.N., Ajayan, P.M., (2005). Super-Compressible Foamlike Carbon Nanotube Films. Science 310, 1307-1310. Cao, G., Chen, X., (2006b). Buckling of single-walled carbon nanotubes upon bending: Molecular dynamics simulations and finite element method. Phys. Rev. B73, 155-435. Chan, S.-P., Yim, W.-L, Gong, X.G., Liu, Z.F., (2003). Carbon Nanotube bundles under high pressure: Transformation to Low-Symmetry Structures. Phys. Rev. B68, 075404. Charlotte, M., Truskinovsky, L., (2002). Linear elastic chain with a hyperpre-stress. J. Mech. Phys. Solids 50(2), 217-251. Charlotte, M., Truskinovsky, L., (2008). Towards multi-scale continuum elasticity theory. Cont. Mech. Therm. 20(3), 133-161. Chesnokov, S.A., Nalimova, V.A., Rinzler, A.G., Smalley, R.E., Fisher, J.E., (1999). Mechanical Energy Storage in Carbon Nanotube Springs. Phys. Rev. Lett. 82(2), 343-346. 22

Daraio, C., Nesterenko, V.F., Aubuchon, J. F., Jin, S., (2004). Dynamic Nano-Fragmentation of Carbon Nanotubes. Nano Lett. 4(10), 1915-1918. Daraio, C., Nesterenko, V.F., Jin, S., (2004). Highly Nonlinear Contact Interaction and Dynamic Energy Dissipation by Forest of Carbon Nanotubes. Appl. Phys. Lett. 85(23), 5724-5727. Daraio, C., Nesterenko, V.F., Jin, S., Wang, W., Rao, A.M., (2006). Impact Response by a Foamlike Forest of Coiled Carbon Nanotubes. J. Appl. Phys. 100(6), 064309 - 064309-4. Deck, C.P., Flowers, J., McKee, G.S.B., Vecchio, K., (2007). Mechanical behavior of ultralong multiwalled carbon nanotube mats. J. Appl. Phys. 101(2), 023512 - 023512-9. Deck, C.P., Vecchio, K.S., (2005). Growth of Well-Aligned Carbon Nanotube Structures in Successive Layers. J. Phys. Chem. B, 109(25), 1235312357. El Sayed, T., Mota, A., Fraternali, F., Ortiz, M., (2008). A variational constitutive model for soft biological tissue. J. Biomech. 41(7), 1458-1466. Falvo, M.R., Clary, G.J., Taylor, R.M., Chi, V., Brooks, F.P. Jr, Washburn, S., Superfine, R., (1997). Bending and buckling of carbon nanotubes under large strain. Nature 389(6651), 582-584. Fraternali, F., Porter, M.A., Daraio, C., (2010). Optimal design of composite granular protectors. Mech. Adv. Mat. Struct. 17(1), 1-19. Gibson, L.J., Ashby, M.F., (1998). Cellular Solids: Structure and Properties. Pergamon Press, Oxford, 1988. Gjerde, K., Kjelstrup-Hansen, J., Clausen, C.H., Teo, K.B.K., Milne, W.I., Rubahn, H.G., Iijima, S., (1991). Helical Microtubules of Graphitic Carbon. Nature 3548(6348), 56-58. Hutchens, S.B., Hall., L.J., Greer, J.R., (2010). In situ Mechanical Testing Reveals Periodic Buckle Nucleation and Propagation in Carbon Nanotube Bundles. Adv. Funct. Mater. 20(14), 2338-2346. Iijima, S., Brabec, C., Maiti, A., Bernholc, J., (1995). Structural Flexibility of Carbon Nanotubes. J. Chem. Phys. 104(5), 2089-2092. 23

Liu, J.Z., Zheng, Q.-S., Wang, L.-F., Jiang, Q., (2005). Mechanical properties of single-walled carbon nanotube bundles as bulk materials. J. Mech. Phys. Solids 53(1), 123-142. Liu, Y., Qian, W., Zhang, Q., Cao, A., Li, Z., Zhou, W., Ma, Y., Wei, F., (2008). Hierarchical Agglomerates of Carbon Nanotubes as High-Pressure Cushions. Nano Lett. 8(5), 1323-1327. Maheshwari, V., Saraf, R., (2008). Tactile devices to sense touch on a par with a human finger. Angewandte Chemie-International edition 47(41), 7808-7826. Majumder, M., Chopra, N., Andrews, R., Hinds, B.J., (2005). Enhanced flow in carbon nanotubes. Nature 438(44). Angewandte ChemieInternational edition 47(41), 7808-7826. Mesarovic, S.Dj., McCarter, C.M., Bahr, D.F., Radhakrishnan, H., Richards, R.F., Richards, C.D., McClain, D., Jiao, J., (2007). Mechanical behavior of a carbon nanotube turf. Scripta Mater. 56(2), 157-160. Misra, A., Greer, J.R., Daraio, C., (2009). Strain rate effects in the mechanical response of polymer anchored carbon nanotube foams. Adv. Mater. 20, 1-5. Nesterenko, V.F.,(2001). Dynamics of Heterogeneous Materials. SpringerVerlag, NY. Oberlin, A., Endo, M., Koyama, T., (1976). Filamentous growth of carbon through benzene decomposition. J. Cryst. Grow. 32(3), 335-349. Ortiz, M., (1999). Plastic yielding as a phase transition. J. Appl. Mech 66, 289-298. Pampolini, G., Del Piero, G., (2008). Strain localization in open-cell polyurethane foams: experiments and theoretical model. J. Mech. Mat. Struct. 3(5), 969-981. Pantano, A., Parks, D.M., Boyce, M.C., (2004). Mechanics of deformation of single- and multi-wall carbon nanotubes. J. Mech. Phys. Solids 52, 789-821.

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Peters, M.J., McNeil, L.E., Lu, J.P., Kahn, D., (2000). Structural phase transition in carbon nanotubes bundles under pressure. Phys. Rev. B, 61(9), 5939-5944. Polizu, S., Savadogo, O., Poulin, P., Yahia, L., (2006). Applications of carbon nanotubes-based biomaterials in biomedical nanotechnology. J. Nanosci. Nanotechnol. 6(7), 1883-1904. Puglisi, G., Truskinovsky, L., (2000). Mechanics of a discrete chain with bi-stable elements. J. Mech. Phys. Solids 48(1), 1-27. Puglisi, G., Truskinovsky, L., (2002). Rate independent hysteresis in a bistable chain. J. Mech. Phys. Solids 50(2), 165-187. Puglisi, G., Truskinovsky, L., (2005). Thermodynamics of rate independent plasticity. J. Mech. Phys. Solids 53, 655-679. Qi, H.J., Teo, K.B.K., Lau, K.K.S., Boyce, M.C., Milne, W.I., Robertson, J., Gleason, K.K., (2003). Determination of mechanical properties of carbon nanotubes and vertically aligned carbon nanotube forests using nanoindentation. J. Mech. Phys. Solids 51(11-12). Radushkevich, L. V. , Lukyanovich, V. M., (1952). O strukture ugleroda, obrazujucegosja pri termiceskom razlozenii okisi ugleroda na zeleznom kontakte (About the structure of carbon formed by thermal decomposition of carbon monoxide on iron substrate). Zurn. Fisic. Chim. 26, 88-95. Suhr, J., Victor, P., Cl, L., Sreekala, S., Zhang, X., Nalamasu, O., Ajayan, P.M., (2007). Fatigue resistance of aligned carbon nanotube arrays under cyclic compression. Nat. Nanotechnol. 2, 417-421. Tao, T., Yang, Z., Delzeit, L., Kashani, A., Meyyappan, M., Majumdar, A., (2008). Height independent compressive modulus of vertically aligned carbon nanotube arrays. Nano Lett. 8(2), 511-515. Teo, E.H.T., Yung, W.K.P., Chua, D.H.C., Tay, B.K., (2007). A carbon nanomattress: a new nanosystem with intrinsic, tunable, damping properties. Adv. Mater. 19(19), 2941-2945. Veedu, V.P., Cao, A., Li, X., Ma, K., Soldano, C., Kar, S., Ajayan, P.M., Ghasemi-Nejhad, M.N., (2006). Multifunctional composites using reinforced laminae with carbon-nanotube forests. Nat. Mater. 5, 457-462. 25

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26

ÈDN È 8

6

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Figure 6: (Color online) Response of uniform chains of microscopic bi-stable springs with properties shown in Tab. 1 of the Appendix: DN dissipation (kJ/m2 ); σ global stress (MPa); ε global strain (filled marks: loading; unfilled marks: unloading).

27

(animation)

Figure 7: (Color online) Selected equilibrium configurations (left) and deformation animation (right - online version only) of the model described in Tab. 1 of the Appendix, for N = 5.

28

Σ@MPaD 10 1st 2nd 3rd 4th

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Figure 8: (Color online) Fitting of compression tests on a doubly anchored CNT foam (top) to non-uniform ‘asymmetric’ spring models (properties in Tab. 2 of the Appendix).

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Figure 9: (Color online) Fitting of experimental compressive stress-strain curves of CNT foams (Cao et al., 2005) to non-uniform ‘asymmetric’ spring models (properties in Tab. 4 of the Appendix).

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Supplementary Self-Built PDF

Supporting online material for Multiscale mass-spring models of carbon nanotube foams F. Fraternali* Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy, and Division of Engineering, King’s College London, Strand, London WC2R 2LS, UK

T. Blesgen Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany

A. Amendola Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy

C. Daraio Graduate Aerospace Laboratories (GALCIT) and Applied Physics, Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA 91125, USA

Abstract We present some additions to Section 5 of the article ‘Multiscale mass-spring models of carbon nanotube foams’, to be published online alongside its electronic version.

Email addresses: [email protected] (F. Fraternali* ), [email protected] (T. Blesgen), [email protected] (A. Amendola), [email protected] (C. Daraio) *Corresponding author

Preprint submitted to Journal Mech. Phys. Solids

September 6, 2010

Addendum to Section 5. Applications Section 5.1. Convergence study and micromechanics of ‘plastic’ steps The material properties used for the convergence study presented in Section 5.1 of the paper are given in Tab. 1 . k0 [MPa] ∆σ/σa 50.00 −0.7

εa εc 0.25 0.77

Table 1: Mechanical properties of uniform chains of microscopic bi-stable springs employed in Section 5.1 of the paper.

Section 5.2. Fitting of experimental results on compressed CNT foams The fits presented in Section 5.2 of the paper consider single loadingunloading cycles, described by data sets of the form n o {εr − ε0 , σ ¯r }r=1,...,Nd , (1) where εr are experimental observations of the global strain ε (hard-device conditions); ε0 is the permanent strain eventually accumulated during a previous load history (mechanical preconditioning); σ ¯r are the experimental recordings of the overall stress σ; and Nd is the number of data points. We look for the best-fit values of the constitutive parameters of model # 2 n o i i i a i i i p= k0 , ∆σ , kc , εi , εc , kh+ kh− i=1,...,N −1 , (2) under simple bounds of the form ub lb ub p ∈ D = [plb 1 , p1 ] × . . . × [pP , pP ],

(3)

where P is the overall number of parameters. We set P = 5N , by prescribing i i i the ratios kh+ /k0 and kh− /k0 . In detail, we fixed kh+ = k0 × 10−3 and i −2 kh− = k0 × 0.5 × 10 for the first cycle of the experiment analyzed in i i Section 5.2.1; kh+ = kh− = k0 × 10−5 for the fourth cycle of the experiment i i analyzed in the same Section; and kh+ = kh− = k0 ×10−2 for the experiments analyzed in Section 5.2.2 The fitting performance of a given set of parameters p was evaluated through the fitting fitness function f (p) =

max |σr (p) − σ ¯r |

r=1,...,Nd

2

(4)

which is the maximum-norm of the piecewise continuous residuals σ−¯ σ . Here, σr (p) denotes the numerically predicted overall stress for ε = εr , coinciding at equilibrium with the stress in each individual spring. The multivariate minimization problem min f (p), (5) p∈D

is expected to be strongly non-convex (Ogden et al., 2004) and well suited for genetic algorithms (Schmitt, 2004; El Sayed et al., 2008). We employed the Breeder Genetic Algorithm (BGA) presented in De Falco et al. (1996) and successfully used as a parameter identification tool in Fraternali et al. (2009). We used a population size of 2P individuals; an initial, randomlychosen, truncation rate equal to 15%, extended intermediate recombination, mutation rate in the interval [10%, 50%], and a maximum number of generations equal to 200. We refer the reader to De Falco et al. (1996) for further technical details of the employed BGA. Section 5.2.1. Compression tests on a doubly anchored CNT foam Tab. 2 illustrates the best fit material parameters obtained for the experiment illustrated in Section 5.2.1 of the paper. Section 5.2.2. Compression tests on a foamlike CNT film We fit the ‘symmetric’ formulation of model # 2 (kci = k0i in each spring) to the first cycle of the cyclic compression test given in Cao et al. (2005), employing chains with 3, 5 and 10 mesoscopic springs. The best fit parameters obtained for this case are given in Tab. 3, and the corresponding stress-strain plots are shown in Fig. 1. One observes that the matching between predictions and experimental recordings appreciably increases by progressively adding mesoscopic springs to the model under consideration (the fitting fitness function f decreases from 1.586 MPa to 1.462 MPa, by letting the number of springs increase from 3 to 10). A good matching between theory and experiments was also observed for what concerns the localization of the CNT deformation during the buckling (‘plastic’) phase. Fig. 2 shows selected equilibrium configurations and a deformation animation of the best fit model with 5 springs. It is worth noting that the succession of the spring snaps depicted in Fig. 2 qualitatively reproduces the progressive kinking of the tubes observed during the test, which is clearly described by Figs. 1, 2 and 4A of Cao et al. (2005). The best fit parameters obtained for ‘asymmetric’ formulation of model # 2, which is presented in Section 5.2.2 of the 3

4 springs, spring # 3 2 1 0 4 springs, spring # 3 2 1 0 4 springs, spring # 3 2 1 0

1. cycle, ε0 k0i [MPa] 11.82 48.89 86.95 16.92 4. cycle, ε0 k0i [MPa] 35.93 10.76 20.42 7.16 4. cycle, ε0 k0i [MPa] 76.29 10.76 1.66 2.24

= 0, f = 0.491 MPa ∆σ i /σai εia εic kci /k0i −0.82 0.09 0.95 9.06 −0.88 0.05 0.93 8.66 −0.75 0.06 0.61 9.58 −0.79 0.27 0.56 9.77 = 0.20, numerical (1), f = 0.484 MPa ∆σ i /σai εia εic kci /k0i −0.69 0.03 0.84 92.73 −0.98 0.88 0.88 59.37 −0.86 0.19 0.27 25.93 −0.16 0.02 0.95 96.70 = 0.20, numerical (2), f = 0.746 MPa ∆σ i /σai εia εic kci /k0i 0.00 0.62 0.62 84.16 0.00 0.88 0.88 59.37 0.00 0.73 0.73 76.13 0.00 0.13 0.72 92.55

Table 2: Mechanical properties of ‘asymmetric’ spring models fitting compression tests on a doubly anchored CNT foam (cf. Section 5.2.1 of the paper).

4

paper, are given in Tab. 4. 3 springs, 1. cycle, f = 1.568 MPa spring # k0i [MPa] ∆σ i /σai εia 2 52.30 −0.78 0.17 1 52.30 −0.78 0.23 0 52.30 −0.78 0.27 5 springs, 1. cycle, f = 1.512 MPa spring # k0i [MPa] ∆σ i /σai εia 4 45.97 −0.92 0.24 3 44.55 −0.76 0.23 2 44.58 −0.78 0.21 1 44.45 −0.76 0.30 0 47.76 −0.73 0.25 10 springs, 1. cycle, f = 1.462 MPa spring # k0i [MPa] ∆σ i /σai εia 9 46.76 −0.81 0.20 8 58.04 −0.86 0.58 7 51.92 −0.80 0.25 6 48.27 −0.88 0.22 5 60.76 −0.86 0.21 4 58.10 −0.72 0.25 3 53.87 −0.70 0.18 2 44.27 −0.66 0.25 1 59.50 −0.64 0.19 0 59.73 −0.76 0.21

εic 0.78 0.88 0.73 εic 0.78 0.72 0.80 0.84 0.83 εic 0.86 0.54 0.87 0.87 0.60 0.81 0.80 0.89 0.64 0.88

Table 3: Mechanical properties of ‘symmetric’ spring models fitting results by Cao et al. (2005).

References Cao, A., Dickrell, P.L., Sawyer, W. G., Ghasemi-Neihad, M.N., Ajayan, P.M., (2005). Super-Compressible Foamlike Carbon Nanotube Films. Science 310, 1307-1310. De Falco, I., Del Balio, R., Della Cioppa, A., Tarantino, E., (1996). A comparative analysis of evolutionary algorithms for function optimization. Pro5

5 springs, 1. cycle, ε0 = 0, f = 0.870 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 4 39.06 −0.69 0.28 0.86 4.70 3 43.41 −0.65 0.31 0.81 4.71 2 48.48 −0.90 0.21 0.81 3.90 1 38.15 −0.73 0.24 0.76 5.00 0 46.89 −0.77 0.26 0.80 4.33 10 springs, 1. cycle, ε0 = 0, f = 0.808 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 9 45.15 −0.70 0.24 0.85 1.85 8 48.16 −0.56 0.30 0.66 5.82 7 39.88 −0.60 0.25 0.72 7.91 6 41.73 −0.61 0.19 0.76 2.70 5 54.97 −0.69 0.21 0.86 7.58 4 59.29 −0.92 0.21 0.86 2.20 3 44.24 −0.89 0.24 0.86 4.97 2 48.96 −0.79 0.27 0.88 9.68 1 61.24 −0.69 0.16 0.83 7.16 0 43.92 −0.67 0.29 0.83 6.69 5 springs, 1000. cycle, ε0 = 0.14, numerical (1), f = 0.444 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 4 8.66 −0.28 0.19 0.90 27.35 3 2.55 −0.27 0.74 0.74 47.35 2 86.24 −0.96 0.59 0.63 49.84 1 57.13 −0.31 0.44 0.60 20.15 0 2.95 −0.71 0.49 0.85 38.13 5 springs, 1000. cycle, ε0 = 0.14, numerical (2), f = 0.926 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 4 4.40 0.00 0.37 0.89 45.19 3 54.24 0.00 0.44 0.71 35.32 2 4.62 0.00 0.59 0.63 26.10 1 1.83 0.00 0.87 0.87 32.26 0 11.38 0.00 0.09 0.75 33.84 Table 4: Mechanical properties of ‘asymmetric’ spring models fitting results by Cao et al. (2005).

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ceedings of the Second Workshop on Evolutionary Computation (WEC2), Nagoya, JAPAN, 29-32. El Sayed, T., Mota, A., Fraternali, F., Ortiz, M., (2008). A variational constitutive model for soft biological tissue. J. Biomech. 41, 1458-1466. Fraternali, F., Porter, M.A., Daraio, C., (2010). Optimal design of composite granular protectors. Mech. Adv. Mat. Struct., 17, 1-19. Ogden, R.W., Saccomandi, G. , Sgura, I., (2004). Fitting hyperelastic model to experimental data. Comput. Mech 34(6), 484-502. Schmitt, L.M.,(2004). Theory of Genetic Algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling. Theor. Comput. Sci. 310(1-3), 181-231.

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Multiscale mass-spring models of carbon nanotube foams F. Fraternali* Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy, and Division of Engineering, King’s College London, Strand, London WC2R 2LS, UK

T. Blesgen Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany

A. Amendola Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy

C. Daraio Graduate Aerospace Laboratories (GALCIT) and Applied Physics, Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA 91125, USA

Abstract This article is concerned with the mechanical properties of dense, vertically aligned carbon nanotube foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition), which represents a finite portion of the foam thickness. An upper Email addresses: [email protected] (F. Fraternali* ), [email protected] (T. Blesgen), [email protected] (A. Amendola), [email protected] (C. Daraio) *Corresponding author Preprint submitted to Journal Mech. Phys. Solids

September 6, 2010

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This article is concerned with the mechanical properties of dense, vertically aligned carbon nanotube foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where in nitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation a non-homogeneous buckling regime, and a densi cation response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain de nes a mesoscopic dissipative element (micro-meso transition), which represents a nite portion of the foam thickness. An upper scale model formed by a chain of nonuniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are tted with excellent agreement. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: a. REPORT

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Graphical Abstract

Graphical abstract

Σ@MPaD 25

1st cycle, 5 springs experimental

20

numerical 15

(movie) 10

5

0

Ε 0

Axial strain localization in a mesoscopic chain of five bistable springs. The spring collapse mimics the local kinking of compressed carbon nanotubes.

0.2

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0.8

Predicted stress-strain response (solid line) at the macroscopic scale, reproducing the experimental behavior of a real CNT foam (dashed line).

Research Highlights

Research highlights • Axial strain localization in microscopic bistable spring chains mimics kinking of compressed carbon nanotube arrays. • Infinitesimal viscous events at the microscale induce time-independent hysteresis at the mesoscale. • Multiscale mechanical modeling of CNT foams is obtained through an informationpassing approach. • Available experimental results on compressed CNT foams are reproduced with excellent agreement.

1

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Multiscale mass-spring models of carbon nanotube foams F. Fraternali* Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy, and Division of Engineering, King’s College London, Strand, London WC2R 2LS, UK

T. Blesgen Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany

A. Amendola Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy

C. Daraio Graduate Aerospace Laboratories (GALCIT) and Applied Physics, Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA 91125, USA

Abstract This article is concerned with the mechanical properties of dense, vertically aligned carbon nanotube foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition), which represents a finite portion of the foam thickness. An upper Email addresses: [email protected] (F. Fraternali* ), [email protected] (T. Blesgen), [email protected] (A. Amendola), [email protected] (C. Daraio) *Corresponding author Preprint submitted to Journal Mech. Phys. Solids

September 6, 2010

scale model formed by a chain of nonuniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are fitted with excellent agreement. Key words: Carbon nanotube foams, bi-stable springs, multiscale behavior, strain localization, hysteresis 1. Introduction Since their discovery (Radushkevich and Lukyanovich, 1952; Oberlin et al., 1976; Iijima et al., 1995), carbon nanotubes (CNTs) have been widely studied for the understanding of different aspects of their chemical, electrical and mechanical responses. Because of their unique properties and their multiscale nature, forests of vertically aligned carpets of CNTs have been proposed for a variety of applications (Gjerde et al., 1991; Veedu et al., 2006; Daraio et al., 2004b; Majumder et al., 2005; Maheshwari and Saraf, 2008). The mechanical response of individual nanotubes under axial and radial deformation, and their bending/buckling modes, have been studied extensively using experimental, theoretical and molecular-dynamics analysis (Iijima et al., 1995; Yakobson et al., 1996; Falvo et al., 1997; Belytschko et al., 2002; Arroyo and Belytschko, 2003; Pantano et al., 2004; Cao and Chen, 2006). The elastic modulus E of individual carbon nanotubes has been reported to be very high: ∼ 1 TPa (Pantano et al., 2004). However in experiments this value can vary widely, depending on the number of defects, the CNT microstructure and the synthesis method followed. The study of the mechanical properties of CNTs was later extended to bundles of nanotubes under pressure (Chesnokov et al., 1999; Peters et al., 2000; Chan et al., 2003; Liu et al., 2005; Qi et al., 2003) and to CNT forests under nanoindentation (Mesarovic et al., 2007). The study of thin structural foams (Gibson and Ashby, 1998) for cushioning (Zhang et al., 2009), energy dissipation (Teo et al., 2007) and protection (Liu et al., 2008) has recently received increasing attention for several practical applications, including mitigation of explosive loading (Nesterenko, 2001). Nanotube based films have been reported as an excellent alternative to regular foams, exhibiting a supercompressible foam-like behavior under compressive cycling loads (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Cao et al., 2005). 2

Investigations on the dynamic response of foam-like forests of CNTs under dynamic ball impacts have also been performed (Daraio et al., 2004b; Daraio et al., 2004b; Daraio et al., 2006; Misra et al., 2009). Results show a strongly nonlinear response that appears to be very suitable for energy-absorbing layered materials in noise and shock wave mitigation and as nonlinear springs for assembling nonlinear acoustic crystals. The CNT forests’ response was also found to be strongly dependent on the forests’ microstructure (height, density, alignment, etc.) and growth method. In certain cases the possible presence of plastic deformation and fracturing of the tubes was reported (Daraio et al., 2004b). Mechanical models consisting of chains of identical bi-stable springs have been extensively studied by several authors, since the can describe a series of relevant nonlinear material behaviors (e.g., phase transformations, reversible pseudo-plasticity, hysteresis, fracture), through the interplay between macroscopic and microscopic length scales (refer e.g. to Ortiz, 1999; Puglisi and Truskinovsky, 2000, 2002, 2005). It is well known that such systems exhibit a ‘bumpy’ multi-well energy landscape, allowing for multiple metastable equilibria (cf., e.g., Blesgen, 2007; Braides and Cicalese, 2007; Charlotte and Truskinovsky, 2002, 2008; Ortiz, 1999; Puglisi and Truskinovsky, 2000, 2002). In particular, Pampolini and Del Piero (2008) have recently found that they well describe the hysteretic response of open-cell polyurethane foams under confined compression tests. In this article, we present a phenomenological model of the mechanical response of carbon nanotube foams under compressive loading, which is inspired by some distinctive features of the micromechanical response reported earlier (cf., e.g., Cao et al., 2005; Zbib et al., 2008; Misra et al., 2009; Hutchens et al., 2010). The given model makes use of multiscale chains of lumped masses connected by nonlinear springs. It captures the ‘three-phase’ compressive deformation response of CNT forests shown by a number of experimental studies. The compressive deformation response is characterized by an initial elastic deformation, a non-homogeneous buckling (or plateau) regime, often featuring a sawtooth-like profile, and a densification phase. This three-phase response, which is common for cellular materials (Gibson and Ashby, 1998), is usually accompanied by marked hysteresis and strain localization in CNT structures. We show in Section 3 that a series of bistable elastic springs (Fig. 1) described by the potential in Eq. (1) exhibits a similar stress-strain response (Fig. 6), and through-the-thickness localization of the axial deformation (Fig. 7). The latter effect mimics the snap-buckling 3

events observed through Scanning Electron Microscope (SEM) in real CNT arrays (cf. Fig. 2 of Cao et al., 2005; Fig. 3 of Hutchens et al., 2010). Such a model therefore appears to be effective in describing the microstructure rearrangements taking place in compressed CNT foams, within a simple 1D framework. We focus on the time-independent component of the hysteresis associated with the compressive loading/unloading of CNT foams, which is essentially due to effects such as friction, entanglement and electrostatic interaction between individual and bundles of carbon nanotubes (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Misra et al., 2009; Cao et al., 2005). The macroscopic hysteresis is described as a rate-independent phenomenon induced by the succession of infinitesimal viscous events at the microscopic scale (Puglisi and Truskinovsky, 2005). As a result, the mechanical model presented in this work does not account for viscosity or other rate-dependent effects at the macroscopic scale, which we address to future work. Most of the available studies on bistable spring chains consider only two spatial-temporal micro-macro scales. The formulation adopted in the present study instead introduces three different time-space scales: a ‘microscopic’ scale (of the order of nanometers), which is associated with the individual bi-stable springs and an infinitesimal portion of the total foam thickness Ltot ; a ‘mesoscopic scale’ (of the order of micrometers), corresponding to the limit of an infinite series of microscopic springs and representing a finite portion of Ltot ; and a ‘macroscopic’ scale (of the order of millimeters), describing the entire foam. We start with the derivation of a discrete 1D model at the microscopic level, showing N + 1 particles having nearest neighbors connected by N uniform nonlinear springs. The discrete spring potentials are chosen to allow the study of softening and strain localization of the tubes. This is discussed in Section 2. Here, we build on the concept of bi-stable springs discussed by Puglisi and Truskinovsky in (Puglisi and Truskinovsky, 2000; Puglisi and Truskinovsky, 2002; Puglisi and Truskinovsky, 2005;). In Section 3 we determine analytically the continuum limit N → ∞ of the microscopic spring chain, by particularizing to the present case the analysis presented in Puglisi and Truskinovsky (2005). Such a limit defines a mesoscopic dissipative element. In Section 4 we then formulate an upper scale model of the entire CNT foam through an information-passing approach, by superimposing a finite number of mesoscopic dissipative springs with different mechanical properties. We account in this way for inhomogeneities 4

induced by the CNT growth process, which several authors think be a leading cause of the discrete folding/buckling events described above (Cao et al., 2005; Hutchens et al., 2010). The introduction of non-uniform mechanical properties allows us to enrich the formulation given in Puglisi and Truskinovsky (2000), Puglisi and Truskinovsky (2002), Puglisi and Truskinovsky (2005), Pampolini and Del Piero, (2008) for bistable spring chains, modeling macroscopic hardening, instead of a perfectly ‘plastic’ response. Hardeningtype post-buckling regimes are commonly observed in compression tests on CNT foams (Cao et al., 2005; Misra et al., 2009). We present in Section 5 a numerical micro-meso convergence study, and the fitting of mesoscopic model parameters to different available experimental data on compressed CNT foams. We demonstrate that the proposed model and its numerical implementation are capable of reproducing the physical behavior of real CNT foams with excellent agreement. We end the article with a critical evaluation and discussion of the results and an outlook. 2. The mechanical model at the microscopic scale We model an infinitesimal portion of a CNT array as a collection of N + 1 lumped masses m0 , . . . , mN piled up one over the other (with N > 2). In this configuration, m0 is clamped at the bottom of the pile at the position x0 = 0, whereas mN is on top at position xN = L > 0, referring to the unstressed reference configuration (Fig. 1c). The nearest neighboring mass points are connected by N nonlinear microscopic springs. We assume that the reference configuration displays equal distances hN := L/N between the masses, and denote the axial displacement of the mass mi (positive upward) by uiN (with u0N = 0, see Fig. 1c). We set uN := {u0N , ..., uN N }. i For the mechanical energy V of the microscopic spring placed between nearest neighbors mi−1 and mi , we assume the three-branch expression defined by i i Va (ε ) = −k0i [εi +ln(1−εi )], εi < εia , i i Vbi (εi ) = c1 + σai εi + 12 kbi (εi −εia )2 , εia ≤ εi ≤ ε¯ic , V (ε ) = i i Vc (ε ) = c2 −kci [εi −εi∗ +ln(1−(εi −εi∗ ))], ε¯ic < εi ,

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(1)

r Figure 1: (Color online) (a) Schematic diagram of a vertically aligned CNTs foam, uniformly loaded in compression. (b) SEM of the as grown carbon nanotube film showing the alignment and the microstructural layering due to the growth process (Deck and Vecchio, 2005). (c) Modeling of a portion of a CNT foam as a collection of microscopic mass-spring elements.

with εi = εi − εi0 , where i

i

ε = ε (uN ) =

i ui−1 N − uN hN

(2)

is the strain measure associated with such a spring (positive in compression). In (1), the quantity εi0 ≥ 0 determines the value of εi corresponding to the first minimum of V i (‘equilibrium’ or ‘initial’ strain); k0i > 0, kbi < 0, kci > 0, εia > 0 and εic ≥ εia are constitutive parameters (five independent parameters); the constants c1 < 0 and c2 > 0 are such that Vai (εia ) = Vbi (εia ), Vbi (¯ εic ) = Vci (¯ εic ); and it results

6

εi∗

=

ε¯ic =

εic

σai − i , kc + σai

(3)

εic (kci + σai ) (σci − σai ) (kci + εic kci + εic σai ) , + kci + σci (kci + σai ) (kci + σci )

(4)

with

σai = k0i

εia , 1 − εia

σci = σai + kbi (¯ εic − εia ).

(5)

The relationship

σ i (εi ) = V i

0

εi εi < εia , k0i 1−ε i, σai + kbi (εi − εia ), εia ≤ εi ≤ ε¯ic , = kci (εi −εi∗ ) , ε¯ic < εi 1−(εi −εi ) ∗

defines the stress σ i acting in the spring connecting mi−1 with mi .

7

(6)

We deduce from (1) that V i owes the two-well profile shown in Fig. 2. Accordingly, the stress-stain relationship (6) is described by the non-monotone profile depicted in Fig. 3. It is worth noting that the mechanical response of the generic microscopic spring encompasses two stable phases, for εi < εia (phase a) and εi > ε¯ic (phase c), respectively; and an intermediate unstable phase b for εia ≥ εi ≥ ε¯ic (spinodal regime). For future use, we set ∆σ i := σci − σai ,

ε¯ia :=

εia σci σai + εia ∆σ i

(7)

i denote the slope of the linear branch of the convex hull of V i and let σM (Maxwell stress, see Fig. 3). We also convene to denote the value of εi where i i in phase c σ i = σM in phase a by εiaM , and the value of εi such that σ i = σM i by εcM . With reference to the generic spring, Equation (6)1 highlights that k0i represents the elastic stiffness at zero stress in phase a (εi = 0). On the other hand, Equation (6)2 points that kbi represents the (constant) stiffness in the spinodal phase b, while Equation (6)3 reveals that kci represents the stiffness at zero stress in phase c (εi = εi∗ ). We name ‘symmetric’ the case with kci = k0i and ‘asymmetric’ the one with kci 6= k0i . The meaning of the other quantities appearing in Equations (1) - (7) is illustrated in Figs. 2 and 3.

3. Dynamic relaxation and hysteresis at the mesoscale There are two fundamental characteristics reported experimentally in the bulk compressive response of dense vertically aligned CNTs forests: (i) a good recovery of deformation even at large compressive strains after a sufficiently large recovery time (Cao et al., 2005; Misra et al., 2009); and (ii) a strong hysteresis, observed in particular in as-grown foams, attributed to several potential effects, including friction, entanglement and electrostatic interaction between individual and bundles of carbon nanotubes (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Misra et al., 2009; Cao et al., 2005). In the present Section we will analyze such a dissipative behavior, by studying a dynamic switching process at the microscopic scale between the phases (a) and (c) described in Figs. 2 and 3. This is in line with the ideas in Puglisi and Truskinovsky (2002), Puglisi and Truskinovsky (2005). Following Puglisi and Truskinovsky (2005), we name a response of the material plastic, if the strain εi of a single spring exceeds εi0 + εia . For 8

Vi

C2

C1 Mc

A1 A2 Εia

Ma ΕiaM Εia

Εic ΕicMΕic

Εi =Εi -Εi0

Figure 2: (Color online) Mechanical energy V i of the generic microscopic (bi-stable) spring.

a chain of N springs, this can be characterized by the occurrence of loading and unloading stress plateaux. However, we notice that our analysis excludes accumulation of permanent deformation, therefore the end point of one hysteresis cycle coincides with the start point of the next cycle. In this sense, the present hysteresis model is time-independent. We will show in Section 5.2 that it is nevertheless capable of capturing a previously accumulated permanent deformation (mechanical preconditioning), through suitable definition of the initial strains εi0 . Within the current Section, we rescale for simplicity L to unity, and assume that V i is independent of the spatial position. Accordingly, we drop the superscript i in front of the spring properties. This because we refer the following analysis to a finite portion of the CNT foam, regarding such a mesoscopic element as the limit for N → ∞ of a series of N identical microscopic springs. Furthermore, we restrict our attention to the case with kc = k0 (”symmetric” model), and ε0 = 0. For the present analysis we 9

Σi

C1

A1 Σai Ma A2

Mc C2

Σ1i =ΣMi

Σci Εia

ΕiaM Εia

Εic ΕicMΕic

Εi =Εi -Εi0

Figure 3: (Color online) Stress σ i versus strain εi relationship in the generic microscopic spring.

require a certain smallness condition on εa and εc relating to strong pinning that disappears in the limit N → ∞, see Puglisi and Truskinovsky (2002). We define the mechanical energy of the foam as N 1 X EN (uN ) = V (εi (uN )) N i=1

(8)

with the effective potential if ε < εa , −k0 [ε + ln(1 − ε)] kb 2 V (ε) = c + σa ε + 2 (ε − εa ) if εa ≤ ε ≤ εc , 1 c2 − k0 [ε − ε∗ + ln(1 − ε + ε∗ )] if εc < ε. Let σ be the given total stress, coinciding at equilibrium with the stress in each individual spring (σ = σ 1 = . . . = σ N ). The total average strain is 10

simply N 1 X i ε (uN ). ε(uN ) := N i=1

We model plasticity by the gradient flow equations (Puglisi and Truskinovsky, 2005) ∂ΦN ν ε˙i (uN ) = − i (ε1 (uN ), . . . , εN (uN )) (9) ∂ε with the total energy N 1 X ΦN (ε , . . . , ε ) := V (εi ) − σεi . N i=1 1

N

The evolution equation (9) lets εi evolve towards local minimizers of ΦN . We are interested in the limit ν → 0 which amounts to infinitely fast evolution such that ε(uN ) attains a local minimizer of ΦN . First we construct the equilibrium points. Inside the i-th spring element, the strain must satisfy the condition V 0 (εi ) = σ. For given total stress σ, there are at most the three local minimizers ε˘a =

σ , k0 + σ

ε˘b =

σ − σa + εa , kb

ε˘c =

σ(1 + ε∗ ) + k0 ε∗ = ε˘a + ε∗ . k0 + σ

Let p, q, 1 − p − q denote the phase fractions of the minimizers a, b, and c, which corresponds to having N p, N q, N (1 − p − q) springs in phase a, b, and c, respectively. As ε 7→ V (ε) is concave in Regime b, if the elongation of a spring in the local minimum ε˘b is altered by an arbitrarily small perturbation, it will move (according to the sign of the perturbation) to either ε˘a or ε˘c . In consequence, any system of N springs with q 6= 0 is unstable and we may in the following restrict to the case q = 0. From ε = p˘ εa +(1−p)˘ εc we compute the equilibrium stress-strain relation to be k0 (ε − εp ) (10) σ(ε) = 1 − (ε − εp ) with εp := (1 − p)ε∗ that can in a natural way be identified with the plastic strain. From (10) we see that σ only depends on the elastic strain εel := ε − εp . 11

For the energy of the equilibrium configuration with N p springs in phase a we find i h σ(ε) k0 bp (ε) = −k0 + ln + (1 − p)c2 E k0 + σ(ε) k0 + σ(ε) = −k0 (ε − εp ) + ln(1 − (ε − εp )) + (1 − p)c2 . bp has a finite number of local minimizers (depending on the Note that E remaining parameter p). As explained in Puglisi and Truskinovsky (2005), the switching takes place between branches that differ in exactly one element in the phase state and the succession of N such steps describes the transition from one homogeneous state to the next. Each of these steps can be thought of as the combination of an elastic part and a plastic part. The stress-strain curve of the foam follows a sawtooth pattern as illustrated in Fig. 4 (left). We denote by A¯i the end point of the i-th branch. ¯ i → A¯i+1 ) where the system The A¯i are the final states of the elastic steps (B remains on the same metastable branch as long as possible. The plastic steps ¯ i ) are characterized by that the total strain is fixed and the system (A¯i → B switches between metastable branches that are neighbors ([N p] = 1, and [Q] generically denotes the jump of a quantity Q). These considerations lead to the representation k0 εa i ¯ , A = εA¯i , 1 − εa

k0 (εa − ε∗ /N ) i ¯ B = εA¯i , , 1 + ε∗ /N − εa

1≤i≤N

in the (ε, σ)-diagram, where i−1 ε∗ . N

εA¯i := εa +

So we can compute that a plastic step is characterized by [ε] = 0,

[N p] = 1,

[σ] = −

k 0 ε∗ , N (1 − εa )(1 + ε∗ /N − εa )

whereas an elastic step fulfills [ε] =

ε∗ , N

[N p] = 0,

[σ] = +

12

k0 ε∗ . N (1 − εa )(1 + ε∗ /N − εa )

(11)

Σ

Σ

A

1

A

2

B

1

B

2

B2 B1 A2 A1 Ε

Ε

Figure 4: Overall stress-stain pattern for a chain with N = 10 microscopic springs (left), and limit response for N → ∞ (right).

The evolution equation (9) lets εi evolve towards local minimizers of ΦN . Now we want to look at the energetics of the plastic and the elastic regime. For an elastic step we have the energy difference bN = k0 ln(1 − εa ) − ln(1 − εa + ε∗ /N ) + k0 ε∗ + c2 . ∆E N In the same spirit, we calculate that for a system with N ≥ 1 springs, the plastic dissipation is DN

i c +k ε 2 0 ∗ := k0 ln(1 − εa + ε∗ /N ) − ln(1 − εa ) − N 1 ε ∗ c2 + k 0 ε ∗ = k0 − for a ξN ∈ (1 − εa , 1 − εa + ε∗ /N ). (12) ξN N N h

Clearly, ξN → 1 − εa for N → ∞. In one hysteresis cycle, there are N loading steps and N steps when the system is unloaded, so we have totally 2N steps that dissipate energy. The total dissipated energy D in a cycle becomes in the limit N → ∞ 2k ε ε 0 ∗ a D = lim 2N DN = − 2c2 . (13) N →∞ 1 − εa 13

We put the modulus here to ensure that the dissipation is positive. The limit stress-stain pattern for N → ∞ is shown in Fig 4. It corresponds to a ”perfectly plastic” behavior with stress plateaux at σ = σa = k0 εa /(1 − εa ) (loading plateau) and σ = σc = σa + ∆σ (unloading plateau). We emphasize again that this ansatz only works for rate-independent plasticity where the energy only depends on start point and end point of the evolution, but not on the evolution path itself. The limit dissipation (13) equals the area enclosed by the limit stress-strain response. As already observed, we confer the behavior shown in Fig. 4 (right) to a mesoscopic spring element, which represents a finite portion of the CNT foam thickness. 4. Multiscale numerical modeling We formulate in the present Section a multiscale numerical model of a nonlinear mass-spring chain, where each spring represents either a microscopic bistable element (cf. Section 2), or a mesoscopic dissipative element of the kind introduced in the previous Section. We introduce two different time scales: an external (slow) time τ ∈ [τ0 , τ1 ] ruling an evolution law of the applied boundary conditions, and an internal (fast) time t ∈ [t0 , t1 ] governing the dynamic relaxation of the system for fixed τ . Depending on the adopted model for the individual springs, we may have the t corresponds to the microscopic timescale (i.e. to the time ruling the microscopic behavior) and τ to the mesoscopic time (micro-meso transition), or, alternatively, that t corresponds to the mesocopic time and τ to the macroscopic time (meso-macro transition). Let us denote a prescribed displacement time-history of the topmost mass N m by δ(τ ). We introduce a discretization {τ1 , . . . , τM } of the loading interval [τ0 , τ1 ] and compute the system response for fixed τ = τk and δ¯ = δ(τk ), through integration with respect to t of the evolution equations mi u¨ˆiN + γ i uˆ˙ iN = σ i+1 − σ i ,

i = 1, . . . , N,

(14)

which generalize the gradient flow equations (9) (Puglisi and Truskinovsky, 2005). In (14), uˆiN = uˆiN (t) denote transient displacement histories of the masses m0 , . . . , mN at fixed τ ; σ i indicates the current stress in the ith spring (it is understood that it results σ i = 0 for i > N ); and γ 1 , . . . , γ N denote damping coefficients. Since we are only interested in the final equilibrium configuration of the transient internal motion, we ‘overdamp’ such a motion down, by introducing 14

fictitious masses and damping factors in (14). In detail, we set the integration i time step ∆t to unity, and introduce fictitious masses mi = αk ∆t2 , with p ki = hN (V i + V i+1 )00 and α ≥ 100. This ensures ∆t ≤ 0.1 mi /k i (i = 1, ..., N ) (Fraternali et al., 2009). Moreover, we let the generic γ i be equal to the ‘critical’ damping defined as follows √ (15) γ i = 2 mi k i The equations of motion (14) are associated with the initial conditions uˆiN (t = t0 ) = (uiN )(k−1) , i = 0, . . . , N − 1; uˆ˙ iN (t = t0 ) = 0,

¯ uˆN N (t = t0 ) = δ

i = 0, . . . , N

(16)

where (uiN )(k−1) (i = 1, . . . , N ) are the displacements of the masses at the external time τ = τk−1 . Equations (14), (16) are numerically integrated through a fourth-order Runge-Kutta integration scheme, up to an internal time t1 such that it results |σ i+1 − σ i | ≤ 10−6 |σ N | for all i ∈ {1, . . . , N − 1}. When the internal equilibrium is reached, we set k ← k + 1 and re-iterate problem (14). For the micro-meso convergence study of Section 5.1, we consider the stress-strain law described by Equation (6) and illustrated in Fig. 3, for each microscopic spring. For the simulation of compression tests on real CNT foams, we instead adopt nonuniform chains of mesoscopic spring characterized by a suitable numerical regularization of the stress-strain pattern shown in Fig. 4. In detail, we introduce a ‘hardening’ type regularization consisting of the following stress-strain law (Fig. 5)

i

σ =

k0i εi σ (a,i) = (1−ε i) , i (εi −εia ), σ (d,i) = σai + kh+

(εi < εia ) and (flag(k−1) 6= c), (εia ≤ εi ≤ εˆic ) and (flag(k−1) = a),

i εia ≤ εi ≤ ε¯ic ) and (flag(k−1) = c), σ (e,i) = σai + ∆σ i + kh− (εi − ε¯ic ), (ˆ kci (εi −εi∗ ) σ (c,i) = 1−(ε (¯ εic < εi ) and (flag(k−1) 6= a), i −εi ) , ∗ (17) 15

where, for each τ = τk (k = 1, . . . , M ), we set a, (εi < εia ) and (flag(k−1) 6= c), c, (¯ εic < εi ) and (flag(k−1) 6= a), = flag(k−1) , ((εia ≤ εi ≤ εˆic ) and (flag(k−1) = a)), or ((ˆ εia ≤ εi ≤ ε¯ic ) and (flag(k−1) = c)).

flag(k)

(18)

i i and kh− in (17) are constitutive The quantities k0i , ∆σ i , kci , εai , εic , kh+ parameters (seven independent parameters), while εˆia and εˆic are computed by solving for εi the equations

σ (a,i) = σ (e,i) ,

σ (c,i) = σ (d,i)

(19)

respectively. One gets back to a perfectly plastic mesoscopic response by i i setting the regularization (‘hardening’) parameters kh+ and kh− equal to i zero. The quantities ε are given by (2) under the replacement of uiN with uˆiN .

Σi C1

A1 Σai Ma

Mc

ΣMi Σci

A2

C2

`i Εa

Εic

ΕiaM Εia

`i ΕicM Εc

Εi =Εi -Εi0

Figure 5: (Color online) Hardening-type regularization of the mesoscopic response.

16

5. Applications The present Section deals with applications of the above numerical model to a convergence study and the fitting of laboratory tests on the cyclic compression of CNT foams. In the first case, series of springs with constitutive equations (6) were considered (model # 1: micro-meso transition), letting the total number of springs of the system increase progressively (micro-meso transition). In the second case, spring chains with constitutive equations (17) were instead examined (model # 2: meso-macro transition). The optimal fit of available experimental stress-strain curves was performed by playing with the constitutive parameters of equations (17) and the number of mesoscopic springs N . Depending on the value of N , a significant number of parameters may need to be identified. This lead us to employ a fitting procedure based on genetic algorithms, which have been proved to be well-suited for multimodal non-convex optimization (cf. e.g. El Sayed et al., 2008; Fraternali et al., 2009). In all the examined examples, we name global strain the quantity ε = (L − `)/L, ` denoting the total deformed length of the chain. 5.1. Convergence study and micromechanics of ‘plastic’ steps We examined uniform chains with fixed length L = 860 µm and increasing number of microscopic springs N , subject to a complete loading-unloading compression cycle up to a global strain ε = 0.85 (as in the experiments examined in Section 5.2.2). We employed model #1 with kc = k0 (‘symmetric’ case); ε0 = 0; and the material properties listed in Tab. 1 of the Appendix for all the springs. Fig. 6 shows the numerically computed dissipation DN versus strain ε and stress σ versus ε responses of such a model for different numbers of springs (N = 5, 10, 15 and 50). The σ-ε plot in Fig. 6 highlights that the global stress-strain response alternates elastic steps and ‘plastic’ jumps of σ at constant ε, oscillating converging to the ‘perfectly plastic’ mesoscopic behavior shown in Fig. 4. The DN -ε plot in Fig. 6 instead shows that DN converges to the limit dissipation defined in equation (13). The above results therefore confirms the theoretical predictions of Section 3. Selected equilibrium configurations and a deformation animation (online version only) of the model with N = 5 springs are given in Fig. 7. One can easily recognize the configurations corresponding to the plastic steps of the microscopic chain, displaying the snap of a single spring and the simultaneous elastic rearrangement of the remaining springs (cf., e.g., configurations 3 and 5 from the left). It is worth noting that the ordering of the springs is of no relevance in the 17

present case. Therefore, the succession of the spring snaps can indifferently proceed from the bottom to the top of the chain, as shown in Fig. 7, from top to bottom, or in random sequence. In a real-world chain of springs, the parameters as k0 or εa would never be perfectly identical and the small deviations of the different springs would break the symmetry and determine the ordering when the springs snap. 5.2. Fitting of experimental results on compressed CNT foams We examined the experimental results of a cyclic compression test on a doubly anchored CNT foam run at the Graduate Aerospace Laboratories of the California Institute of Technology, and the results given in Cao et al. (2005) on the cyclic compression of a foam-like CNT film. We fit model # 2 both to he first loading/unloading cycle of the examined foams, and to selected cycles following an initial mechanical preconditioning, employing the Breeder Genetic Algorithm (BGA) presented in the Appendix. For the fit of the first cycles we assumed εi0 = 0 in each spring, while for the cycles following the foam preconditioning we set εi0 = ε0 in each spring, ε0 being the macroscopic (permanent or transient) strain observed at zero stress at the end of the previous loading cycle. For the cycles following the preconditioning, we considered two different fitting models: one accounting for hysteresis (numerical (1): ∆σ i < 0 in each mesoscopic spring), and the other without hysteresis (numerical (2): ∆σ i = 0 in each mesoscopic spring). Since all the adopted fitting models account for macroscopic time-independent behavior (cf. Sect. 3), our fitting results are independent of the strain rate actually applied in the experiments. 5.2.1. Compression tests on a doubly anchored CNT foam We performed cyclic compression tests on vertically aligned multi-walled carbon nanotube forests (800µm in length with sample area ∼ 14mm2 ) grown by chemical vapor deposition (CVD) using ferrocene and toluene as precursors. The average diameter of the as-grown CNTs was ∼ 50nm. The as-grown CNTs foams were partially anchored between two polymer layers to provide structural support and sample transportability. For anchoring the CNT foams on a substrate we spin-coated polydimethylsiloxane (PDMS) on top of a glass slide at 800 RPM to get 50 − 100µm thick films. Before curing the polymer we partially embedded the CNT-foams at 80◦ C for 1 hour. After curing, the carbon nanotubes protruding from the substrate showed excellent vertical alignment with an average height of 18

∼ 750µm. The process was repeated turning the sample upside down to obtain a ‘doubly anchored’ system (i.e. sandwich structure with polymer on both sides and the CNT foam in the middle). Typical experimental results obtained from cyclic compression tests are shown in Fig. 8 (top). We separately fit the first and the fourth stress-strain cycles shown in Fig. 8 to model # 2, considering the general ‘asymmetric’ case with kci 6= k0i (i = 0, . . . , N − 1) and a chain with N = 4 springs. For the fourth cycle we accounted for an initial strain ε0 = 0.20, which approximatively corresponds to the strain measured at zero stress at the end of the third loading cycle (Fig. 8, top). The best fit parameters obtained through BGA optimization are given in Tab. 2 of the Appendix. A comparison between best-fit and experimental overall stress-strain curves is shown in Fig. 8 (center and bottom). One observes that the non-uniform dissipative mass-spring model is able to capture the real hysteretical behavior of the examined CNT foam, both at their pristine state (first cycle) and after mechanical preconditioning (fourth cycle), through a multi-plateaux overall stress-strain profile. In particular, Fig. 8-bottom and Tab. 2 of the Appendix show that the response after preconditioning can be roughly described through a suitable non-dissipative model. 5.2.2. Compression tests on a foamlike CNT film We fit a cyclic compression test given in Cao et al. (2005) for a CNT foamlike film with thickness L = 860 µm. The analyzed experiment performed 1000 loading/unloading cycles up to a global strain ε = 0.85 (cf. Fig. 3B of Cao et al., 2005 and Fig. 3 of the Appendix). The fitting of the ‘symmetric’ formulation of model # 2 (kci = k0i in each spring) to the first cycle is presented in the Appendix. A dramatic improvement of the fitting ability of model # 2 was obtained by considering the ‘asymmetric’ case (kci 6= k0i ). As shown in Tab. 4 of the Appendix, we were indeed able to reduce the fitting fitness to 0.808 MPa by using a BGA-optimized ”asymmetric” multiscale model with 10 springs. Such a fitness value is markedly lower than that obtained for the corresponding ”symmetric” case (1.462 MPa, cf. Tab. 3 of the the Appendix). We were able to optimally fit also the 1000th cycle of the examined experiment to the ‘asymmetric’ model, obtaining a fitting fitness of 0.444 MPa through the BGA-optimized 5 spring model described in Tab. 4 of the Appendix. The excellent match between numerical and experimental results for the present case is illustrated by Fig. 9, which compares predicted and measured overall stress-strain responses. For the first cycle 19

we observe that the numerical multi-plateaux response closely approaches the continuous experimental recording as the number of mesoscopic springs increases from 5 to 10 (Fig. 9, top and center). Regarding the 1000th cycle, we assumed ε0 = 0.14 as suggested in Cao et al. (2005). Fig. 9, bottom shows that the experimental stress-strain profile corresponding to the 1000th cycle is already excellently approximated by a 5 spring model with dissipation (numerical (1)), and roughly described by an analogous model without dissipation (numerical (2)). One observes from Tabs. 3 and 4 of the Appendix that the average value of the stiffness k0i , among all the springs, is approximatively equal to the elastic modulus estimated by Cao et al. (2005) for the present CNT foam (≈ 50 MPa). 6. Discussion and Outlook In this article we derived a mechanical model describing the behavior of CNT foams under uniform compression, through the concept of bi-stable springs presented in Puglisi and Truskinovsky (2000, 2002, 2005), which has been recently applied to open-cell polyurethane foams (Pampolini and Del Piero, 2008). The proposed model differs from other bi-stable mass-spring models available in the literature (refer e.g. to Blesgen, 2007; Braides and Cicalese, 2007; Charlotte and Truskinovsky, 2002, 2008; Ortiz, 1999; Puglisi and Truskinovsky, 2000, 2002, 2005, and references therein), due to the presence of an intermediate, mesoscopic scale, placed in between the microscopic scale of the bi-stable springs, and the macroscopic scale of the entire structure. The microscopic scale aims to describe the dynamic snapping of the carbon nanotubes, due to local buckling (Cao et al., 2005; Zbib et al., 2008; Misra et al., 2009; Hutchens et al., 2010), while the mesoscopic scale is intended to describe the time-independent hysteretic behavior of finite portions of the CNT foam. The latter typically follows from kinking, entanglement of the tubes, friction and other microscopic dissipative effects (Suhr et al., 2007; Teo et al., 2007; Tao et al., 2008; Deck et al., 2007; Misra et al., 2009). Another relevant difference between the present model and most of the available bi-stable mass-spring models consists of the fact that the current model accounts for nonuniform mesoscopic spring properties, while other models usually consider uniform chains of bi-stable springs (see e.g. Puglisi and Truskinovsky, 2000, 2002; Pampolini and Del Piero, 2008). Such a mechanical inhomogeneity allows us to account for hardening of the macroscopic 20

response in the post-buckling range (Cao et al., 2005; Misra et al., 2009). The numerical results given in Section 5 demonstrated that our methods are capable of recovering available experimental results on the compression of CNT foams with excellent agreement. We fitted the parameters of the mesoscopic springs through a Breeder Genetic Algorithm, which is well suited for global non-convex optimization. The fitting of experimental stress-strain curves allowed us to recognize that non-uniform dissipative mass-spring models are well suited to capture the main features of the real compressive response of CNT foams, and specifically strain localization due to CNT kinking and time-independent hysteresis. The latter was found to be relevant during loading/unloading from the pristine state, and progressively decaying after mechanical preconditioning. By setting to zero the dissipation of the model, we were led to obtain a rough, non-linearly elastic approximation of the examined experimental behaviors after preconditioning. The distinctive features of the multiscale model presented in this work highlight its potential use to describe the mechanical behavior of multilayered systems composed of alternating CNT foams and anchoring polymeric films (Misra et al., 2009). We will address such an extension of the current model in future work. Other future directions of the present research might regard the description of permanent deformation through fatigue or damage mechanisms; the inclusion of rate-dependent dissipative effects at the macroscopic scale; and the modeling of long-range interactions mimicking van der Waals forces. Acknowledgements The authors would like to thank Dr. Abha Misra and Dr. Luigi de Nardo for performing the experiments presented in Section 5.2.1. CD acknowledges support from the Institute for Collaborative Biotechnologies under contract W911NF-09-D-0001 with the Army Research Office. FF greatly acknowledges the support of the University Centre for Risk Prediction and Prevention (CUGRI) in association between the Universities of Salerno and Napoli Federico II, Italy. FF also thanks the Graduate Aerospace Laboratory at the California Institute of Technology (GALCIT) for the hospitality during his visit.

21

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Figure 7: (Color online) Selected equilibrium configurations (left) and deformation animation (right - online version only) of the model described in Tab. 1 of the Appendix, for N = 5.

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Supplementary Self-Built PDF

Supporting online material for Multiscale mass-spring models of carbon nanotube foams F. Fraternali* Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy, and Division of Engineering, King’s College London, Strand, London WC2R 2LS, UK

T. Blesgen Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany

A. Amendola Department of Civil Engineering, University of Salerno, 84084 Fisciano(SA), Italy

C. Daraio Graduate Aerospace Laboratories (GALCIT) and Applied Physics, Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA 91125, USA

Abstract We present some additions to Section 5 of the article ‘Multiscale mass-spring models of carbon nanotube foams’, to be published online alongside its electronic version.

Email addresses: [email protected] (F. Fraternali* ), [email protected] (T. Blesgen), [email protected] (A. Amendola), [email protected] (C. Daraio) *Corresponding author

Preprint submitted to Journal Mech. Phys. Solids

September 6, 2010

Addendum to Section 5. Applications Section 5.1. Convergence study and micromechanics of ‘plastic’ steps The material properties used for the convergence study presented in Section 5.1 of the paper are given in Tab. 1 . k0 [MPa] ∆σ/σa 50.00 −0.7

εa εc 0.25 0.77

Table 1: Mechanical properties of uniform chains of microscopic bi-stable springs employed in Section 5.1 of the paper.

Section 5.2. Fitting of experimental results on compressed CNT foams The fits presented in Section 5.2 of the paper consider single loadingunloading cycles, described by data sets of the form n o {εr − ε0 , σ ¯r }r=1,...,Nd , (1) where εr are experimental observations of the global strain ε (hard-device conditions); ε0 is the permanent strain eventually accumulated during a previous load history (mechanical preconditioning); σ ¯r are the experimental recordings of the overall stress σ; and Nd is the number of data points. We look for the best-fit values of the constitutive parameters of model # 2 n o i i i a i i i p= k0 , ∆σ , kc , εi , εc , kh+ kh− i=1,...,N −1 , (2) under simple bounds of the form ub lb ub p ∈ D = [plb 1 , p1 ] × . . . × [pP , pP ],

(3)

where P is the overall number of parameters. We set P = 5N , by prescribing i i i the ratios kh+ /k0 and kh− /k0 . In detail, we fixed kh+ = k0 × 10−3 and i −2 kh− = k0 × 0.5 × 10 for the first cycle of the experiment analyzed in i i Section 5.2.1; kh+ = kh− = k0 × 10−5 for the fourth cycle of the experiment i i analyzed in the same Section; and kh+ = kh− = k0 ×10−2 for the experiments analyzed in Section 5.2.2 The fitting performance of a given set of parameters p was evaluated through the fitting fitness function f (p) =

max |σr (p) − σ ¯r |

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which is the maximum-norm of the piecewise continuous residuals σ−¯ σ . Here, σr (p) denotes the numerically predicted overall stress for ε = εr , coinciding at equilibrium with the stress in each individual spring. The multivariate minimization problem min f (p), (5) p∈D

is expected to be strongly non-convex (Ogden et al., 2004) and well suited for genetic algorithms (Schmitt, 2004; El Sayed et al., 2008). We employed the Breeder Genetic Algorithm (BGA) presented in De Falco et al. (1996) and successfully used as a parameter identification tool in Fraternali et al. (2009). We used a population size of 2P individuals; an initial, randomlychosen, truncation rate equal to 15%, extended intermediate recombination, mutation rate in the interval [10%, 50%], and a maximum number of generations equal to 200. We refer the reader to De Falco et al. (1996) for further technical details of the employed BGA. Section 5.2.1. Compression tests on a doubly anchored CNT foam Tab. 2 illustrates the best fit material parameters obtained for the experiment illustrated in Section 5.2.1 of the paper. Section 5.2.2. Compression tests on a foamlike CNT film We fit the ‘symmetric’ formulation of model # 2 (kci = k0i in each spring) to the first cycle of the cyclic compression test given in Cao et al. (2005), employing chains with 3, 5 and 10 mesoscopic springs. The best fit parameters obtained for this case are given in Tab. 3, and the corresponding stress-strain plots are shown in Fig. 1. One observes that the matching between predictions and experimental recordings appreciably increases by progressively adding mesoscopic springs to the model under consideration (the fitting fitness function f decreases from 1.586 MPa to 1.462 MPa, by letting the number of springs increase from 3 to 10). A good matching between theory and experiments was also observed for what concerns the localization of the CNT deformation during the buckling (‘plastic’) phase. Fig. 2 shows selected equilibrium configurations and a deformation animation of the best fit model with 5 springs. It is worth noting that the succession of the spring snaps depicted in Fig. 2 qualitatively reproduces the progressive kinking of the tubes observed during the test, which is clearly described by Figs. 1, 2 and 4A of Cao et al. (2005). The best fit parameters obtained for ‘asymmetric’ formulation of model # 2, which is presented in Section 5.2.2 of the 3

4 springs, spring # 3 2 1 0 4 springs, spring # 3 2 1 0 4 springs, spring # 3 2 1 0

1. cycle, ε0 k0i [MPa] 11.82 48.89 86.95 16.92 4. cycle, ε0 k0i [MPa] 35.93 10.76 20.42 7.16 4. cycle, ε0 k0i [MPa] 76.29 10.76 1.66 2.24

= 0, f = 0.491 MPa ∆σ i /σai εia εic kci /k0i −0.82 0.09 0.95 9.06 −0.88 0.05 0.93 8.66 −0.75 0.06 0.61 9.58 −0.79 0.27 0.56 9.77 = 0.20, numerical (1), f = 0.484 MPa ∆σ i /σai εia εic kci /k0i −0.69 0.03 0.84 92.73 −0.98 0.88 0.88 59.37 −0.86 0.19 0.27 25.93 −0.16 0.02 0.95 96.70 = 0.20, numerical (2), f = 0.746 MPa ∆σ i /σai εia εic kci /k0i 0.00 0.62 0.62 84.16 0.00 0.88 0.88 59.37 0.00 0.73 0.73 76.13 0.00 0.13 0.72 92.55

Table 2: Mechanical properties of ‘asymmetric’ spring models fitting compression tests on a doubly anchored CNT foam (cf. Section 5.2.1 of the paper).

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paper, are given in Tab. 4. 3 springs, 1. cycle, f = 1.568 MPa spring # k0i [MPa] ∆σ i /σai εia 2 52.30 −0.78 0.17 1 52.30 −0.78 0.23 0 52.30 −0.78 0.27 5 springs, 1. cycle, f = 1.512 MPa spring # k0i [MPa] ∆σ i /σai εia 4 45.97 −0.92 0.24 3 44.55 −0.76 0.23 2 44.58 −0.78 0.21 1 44.45 −0.76 0.30 0 47.76 −0.73 0.25 10 springs, 1. cycle, f = 1.462 MPa spring # k0i [MPa] ∆σ i /σai εia 9 46.76 −0.81 0.20 8 58.04 −0.86 0.58 7 51.92 −0.80 0.25 6 48.27 −0.88 0.22 5 60.76 −0.86 0.21 4 58.10 −0.72 0.25 3 53.87 −0.70 0.18 2 44.27 −0.66 0.25 1 59.50 −0.64 0.19 0 59.73 −0.76 0.21

εic 0.78 0.88 0.73 εic 0.78 0.72 0.80 0.84 0.83 εic 0.86 0.54 0.87 0.87 0.60 0.81 0.80 0.89 0.64 0.88

Table 3: Mechanical properties of ‘symmetric’ spring models fitting results by Cao et al. (2005).

References Cao, A., Dickrell, P.L., Sawyer, W. G., Ghasemi-Neihad, M.N., Ajayan, P.M., (2005). Super-Compressible Foamlike Carbon Nanotube Films. Science 310, 1307-1310. De Falco, I., Del Balio, R., Della Cioppa, A., Tarantino, E., (1996). A comparative analysis of evolutionary algorithms for function optimization. Pro5

5 springs, 1. cycle, ε0 = 0, f = 0.870 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 4 39.06 −0.69 0.28 0.86 4.70 3 43.41 −0.65 0.31 0.81 4.71 2 48.48 −0.90 0.21 0.81 3.90 1 38.15 −0.73 0.24 0.76 5.00 0 46.89 −0.77 0.26 0.80 4.33 10 springs, 1. cycle, ε0 = 0, f = 0.808 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 9 45.15 −0.70 0.24 0.85 1.85 8 48.16 −0.56 0.30 0.66 5.82 7 39.88 −0.60 0.25 0.72 7.91 6 41.73 −0.61 0.19 0.76 2.70 5 54.97 −0.69 0.21 0.86 7.58 4 59.29 −0.92 0.21 0.86 2.20 3 44.24 −0.89 0.24 0.86 4.97 2 48.96 −0.79 0.27 0.88 9.68 1 61.24 −0.69 0.16 0.83 7.16 0 43.92 −0.67 0.29 0.83 6.69 5 springs, 1000. cycle, ε0 = 0.14, numerical (1), f = 0.444 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 4 8.66 −0.28 0.19 0.90 27.35 3 2.55 −0.27 0.74 0.74 47.35 2 86.24 −0.96 0.59 0.63 49.84 1 57.13 −0.31 0.44 0.60 20.15 0 2.95 −0.71 0.49 0.85 38.13 5 springs, 1000. cycle, ε0 = 0.14, numerical (2), f = 0.926 MPa spring # k0i [MPa] ∆σ i /σai εia εic kci /k0i 4 4.40 0.00 0.37 0.89 45.19 3 54.24 0.00 0.44 0.71 35.32 2 4.62 0.00 0.59 0.63 26.10 1 1.83 0.00 0.87 0.87 32.26 0 11.38 0.00 0.09 0.75 33.84 Table 4: Mechanical properties of ‘asymmetric’ spring models fitting results by Cao et al. (2005).

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ceedings of the Second Workshop on Evolutionary Computation (WEC2), Nagoya, JAPAN, 29-32. El Sayed, T., Mota, A., Fraternali, F., Ortiz, M., (2008). A variational constitutive model for soft biological tissue. J. Biomech. 41, 1458-1466. Fraternali, F., Porter, M.A., Daraio, C., (2010). Optimal design of composite granular protectors. Mech. Adv. Mat. Struct., 17, 1-19. Ogden, R.W., Saccomandi, G. , Sgura, I., (2004). Fitting hyperelastic model to experimental data. Comput. Mech 34(6), 484-502. Schmitt, L.M.,(2004). Theory of Genetic Algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling. Theor. Comput. Sci. 310(1-3), 181-231.

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Figure 2: Selected equilibrium configurations (left) and deformation animation (right) of the model with five springs described in Tab. 3.

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